| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 2·7-s + 3·8-s + 9-s − 10-s + 2·11-s + 12-s + 2·13-s − 2·14-s − 15-s − 16-s − 2·17-s − 18-s − 20-s − 2·21-s − 2·22-s + 23-s − 3·24-s + 25-s − 2·26-s − 27-s − 2·28-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.554·13-s − 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.426·22-s + 0.208·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.377·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.943439843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943439843\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.73161124518710, −12.29694270888813, −11.61536930977490, −11.15869499168237, −10.89370480467101, −10.53198074897291, −9.758429745808789, −9.570632410761243, −9.064807602539799, −8.620375368157662, −8.111704355932469, −7.751999617241869, −7.116157597901042, −6.642257257288559, −6.166534745566096, −5.476478231306028, −5.244066874303026, −4.558557491966571, −4.037757852195217, −3.861464885007545, −2.721575719650501, −2.219665833342583, −1.457256612465949, −1.051980766155331, −0.5305139344184917,
0.5305139344184917, 1.051980766155331, 1.457256612465949, 2.219665833342583, 2.721575719650501, 3.861464885007545, 4.037757852195217, 4.558557491966571, 5.244066874303026, 5.476478231306028, 6.166534745566096, 6.642257257288559, 7.116157597901042, 7.751999617241869, 8.111704355932469, 8.620375368157662, 9.064807602539799, 9.570632410761243, 9.758429745808789, 10.53198074897291, 10.89370480467101, 11.15869499168237, 11.61536930977490, 12.29694270888813, 12.73161124518710
