| L(s) = 1 | + 2.61·2-s + 3-s + 4.85·4-s + 2.61·6-s − 7-s + 7.47·8-s + 9-s − 5.47·11-s + 4.85·12-s + 0.763·13-s − 2.61·14-s + 9.85·16-s − 7.70·17-s + 2.61·18-s − 3.23·19-s − 21-s − 14.3·22-s + 5·23-s + 7.47·24-s + 2·26-s + 27-s − 4.85·28-s + 4.70·29-s + 4.47·31-s + 10.8·32-s − 5.47·33-s − 20.1·34-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.42·4-s + 1.06·6-s − 0.377·7-s + 2.64·8-s + 0.333·9-s − 1.64·11-s + 1.40·12-s + 0.211·13-s − 0.699·14-s + 2.46·16-s − 1.86·17-s + 0.617·18-s − 0.742·19-s − 0.218·21-s − 3.05·22-s + 1.04·23-s + 1.52·24-s + 0.392·26-s + 0.192·27-s − 0.917·28-s + 0.874·29-s + 0.803·31-s + 1.91·32-s − 0.952·33-s − 3.46·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.472028629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.472028629\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 8.23T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 0.763T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 + 0.527T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 5.70T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−10.88305000630002915801675779947, −10.51598250423239454545138722393, −8.994117574786019984819193779230, −7.894209300014456103447531603063, −6.89146884514217468798889316367, −6.13063408322345187956609937524, −4.93981419341911702196571857375, −4.27364598256038389131842168876, −2.95196754636197338202645198661, −2.34586971382930296741322120989,
2.34586971382930296741322120989, 2.95196754636197338202645198661, 4.27364598256038389131842168876, 4.93981419341911702196571857375, 6.13063408322345187956609937524, 6.89146884514217468798889316367, 7.894209300014456103447531603063, 8.994117574786019984819193779230, 10.51598250423239454545138722393, 10.88305000630002915801675779947
