| L(s) = 1 | − 4-s − 5·5-s − 2·7-s − 4·13-s + 16-s + 5·20-s − 3·23-s + 9·25-s + 2·28-s + 3·29-s + 10·35-s − 7·49-s + 4·52-s − 4·59-s − 64-s + 20·65-s − 17·67-s − 24·71-s − 5·80-s − 9·81-s − 14·83-s + 8·91-s + 3·92-s − 9·100-s + 103-s + 7·107-s − 23·109-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.23·5-s − 0.755·7-s − 1.10·13-s + 1/4·16-s + 1.11·20-s − 0.625·23-s + 9/5·25-s + 0.377·28-s + 0.557·29-s + 1.69·35-s − 49-s + 0.554·52-s − 0.520·59-s − 1/8·64-s + 2.48·65-s − 2.07·67-s − 2.84·71-s − 0.559·80-s − 81-s − 1.53·83-s + 0.838·91-s + 0.312·92-s − 0.899·100-s + 0.0985·103-s + 0.676·107-s − 2.20·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 568516 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 568516 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 29 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 188 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.999024748400384149863581570101, −7.57028456498352388210615312689, −7.21486087086800952250724490362, −6.86895177176343070022073872103, −6.07329823343504802417728791970, −5.79129799738908828318910518445, −4.87230146213429929011666034996, −4.59610082918336053236531092115, −4.11604366843138090401725883678, −3.71522133765283824368225454056, −3.07592587029059791566962296840, −2.70033271275438962230640337986, −1.45991438604170319429892654204, 0, 0,
1.45991438604170319429892654204, 2.70033271275438962230640337986, 3.07592587029059791566962296840, 3.71522133765283824368225454056, 4.11604366843138090401725883678, 4.59610082918336053236531092115, 4.87230146213429929011666034996, 5.79129799738908828318910518445, 6.07329823343504802417728791970, 6.86895177176343070022073872103, 7.21486087086800952250724490362, 7.57028456498352388210615312689, 7.999024748400384149863581570101
