| L(s) = 1 | + 2-s + 4-s − 2·5-s − 0.561·7-s + 8-s − 2·10-s − 11-s + 0.561·13-s − 0.561·14-s + 16-s + 0.561·17-s − 19-s − 2·20-s − 22-s − 1.43·23-s − 25-s + 0.561·26-s − 0.561·28-s + 5.68·29-s + 2·31-s + 32-s + 0.561·34-s + 1.12·35-s − 5.12·37-s − 38-s − 2·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.894·5-s − 0.212·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s + 0.155·13-s − 0.150·14-s + 0.250·16-s + 0.136·17-s − 0.229·19-s − 0.447·20-s − 0.213·22-s − 0.299·23-s − 0.200·25-s + 0.110·26-s − 0.106·28-s + 1.05·29-s + 0.359·31-s + 0.176·32-s + 0.0963·34-s + 0.189·35-s − 0.842·37-s − 0.162·38-s − 0.316·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 - 0.561T + 17T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 0.876T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−8.124004011098077547221534479895, −7.33151835395464245566134753164, −6.62023685926207321505333208385, −5.89946245015647303281298097223, −4.95769548942165425440900356970, −4.31864329565482504465931305608, −3.48600549216742341278472626342, −2.80830678216946578285391133527, −1.55993945373214683109310539871, 0,
1.55993945373214683109310539871, 2.80830678216946578285391133527, 3.48600549216742341278472626342, 4.31864329565482504465931305608, 4.95769548942165425440900356970, 5.89946245015647303281298097223, 6.62023685926207321505333208385, 7.33151835395464245566134753164, 8.124004011098077547221534479895
