| L(s) = 1 | − 2-s + 2.56·3-s + 4-s + 2·5-s − 2.56·6-s − 0.561·7-s − 8-s + 3.56·9-s − 2·10-s + 11-s + 2.56·12-s + 0.561·13-s + 0.561·14-s + 5.12·15-s + 16-s − 0.561·17-s − 3.56·18-s − 19-s + 2·20-s − 1.43·21-s − 22-s + 1.43·23-s − 2.56·24-s − 25-s − 0.561·26-s + 1.43·27-s − 0.561·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.47·3-s + 0.5·4-s + 0.894·5-s − 1.04·6-s − 0.212·7-s − 0.353·8-s + 1.18·9-s − 0.632·10-s + 0.301·11-s + 0.739·12-s + 0.155·13-s + 0.150·14-s + 1.32·15-s + 0.250·16-s − 0.136·17-s − 0.839·18-s − 0.229·19-s + 0.447·20-s − 0.313·21-s − 0.213·22-s + 0.299·23-s − 0.522·24-s − 0.200·25-s − 0.110·26-s + 0.276·27-s − 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.808717920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.808717920\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 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
−10.86878081460772726630137738357, −9.938850994382666261086169202310, −9.269548033660763706910973034590, −8.700309598822903808568055572210, −7.74131100109873532120571409940, −6.79288055299980896065481533675, −5.65341339453139235578459003044, −3.91222257259820018576286398452, −2.72121495121319900109329876429, −1.72927659535003427204067869921,
1.72927659535003427204067869921, 2.72121495121319900109329876429, 3.91222257259820018576286398452, 5.65341339453139235578459003044, 6.79288055299980896065481533675, 7.74131100109873532120571409940, 8.700309598822903808568055572210, 9.269548033660763706910973034590, 9.938850994382666261086169202310, 10.86878081460772726630137738357
