| L(s) = 1 | + 8·2-s − 27·3-s − 165·5-s − 216·6-s − 343·7-s − 512·8-s − 1.32e3·10-s + 2.05e3·11-s + 5.44e3·13-s − 2.74e3·14-s + 4.45e3·15-s − 4.09e3·16-s + 1.57e4·17-s − 6.75e3·19-s + 9.26e3·21-s + 1.64e4·22-s − 3.08e4·23-s + 1.38e4·24-s + 7.81e4·25-s + 4.35e4·26-s + 1.96e4·27-s − 2.36e5·29-s + 3.56e4·30-s + 1.47e5·31-s − 5.54e4·33-s + 1.25e5·34-s + 5.65e4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.590·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 0.417·10-s + 0.465·11-s + 0.686·13-s − 0.267·14-s + 0.340·15-s − 1/4·16-s + 0.775·17-s − 0.225·19-s + 0.218·21-s + 0.329·22-s − 0.528·23-s + 0.204·24-s + 25-s + 0.485·26-s + 0.192·27-s − 1.80·29-s + 0.240·30-s + 0.889·31-s − 0.268·33-s + 0.548·34-s + 0.223·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.457475682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457475682\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{3} T + p^{6} T^{2} \) |
| 7 | $C_2$ | \( 1 + p^{3} T + p^{7} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 33 p T - 2036 p^{2} T^{2} + 33 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2055 T - 15264146 T^{2} - 2055 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2720 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 924 p T - 566081 p^{2} T^{2} - 924 p^{8} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6752 T - 848282235 T^{2} + 6752 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 30828 T - 2454459863 T^{2} + 30828 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 118305 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 147517 T - 5751348822 T^{2} - 147517 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 311732 T + 2244962691 T^{2} + 311732 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 491400 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 577174 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 854862 T + 224165918581 T^{2} - 854862 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1166883 T + 186904795852 T^{2} + 1166883 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 167079 T - 2460736092578 T^{2} - 167079 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1027274 T - 2087450964945 T^{2} + 1027274 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3268114 T + 4619857511673 T^{2} - 3268114 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3046842 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3209110 T - 749011526997 T^{2} - 3209110 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 7127987 T + 31604289686010 T^{2} + 7127987 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4365909 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 11996214 T + 99677815438267 T^{2} - 11996214 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13343639 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.63460853519336747345915063399, −14.35956752108247758006766651906, −13.35842139622683189020505369201, −13.25183383162241060069696753523, −12.21107406768645638860953207956, −12.05238307754813819393408445029, −11.38434376859982458000970198995, −10.73000516771251285944561955742, −10.03595748065803961995119923799, −9.280408766548185635345851696109, −8.480986537180081046464933644006, −7.87556284674725819480913995692, −6.74389955460034718977052555471, −6.41249211535023715291373309356, −5.39526942369842875096236777280, −4.85313812407161934243320415275, −3.65244400963995604798404027332, −3.39755947573295118365560597523, −1.74685286124581370575489214736, −0.47963762385012993818302426908,
0.47963762385012993818302426908, 1.74685286124581370575489214736, 3.39755947573295118365560597523, 3.65244400963995604798404027332, 4.85313812407161934243320415275, 5.39526942369842875096236777280, 6.41249211535023715291373309356, 6.74389955460034718977052555471, 7.87556284674725819480913995692, 8.480986537180081046464933644006, 9.280408766548185635345851696109, 10.03595748065803961995119923799, 10.73000516771251285944561955742, 11.38434376859982458000970198995, 12.05238307754813819393408445029, 12.21107406768645638860953207956, 13.25183383162241060069696753523, 13.35842139622683189020505369201, 14.35956752108247758006766651906, 14.63460853519336747345915063399
