| L(s) = 1 | + 2·2-s + 7·3-s + 4·4-s + 14·6-s − 14·7-s + 8·8-s + 22·9-s + 11·11-s + 28·12-s + 72·13-s − 28·14-s + 16·16-s + 46·17-s + 44·18-s − 20·19-s − 98·21-s + 22·22-s + 107·23-s + 56·24-s + 144·26-s − 35·27-s − 56·28-s + 120·29-s + 117·31-s + 32·32-s + 77·33-s + 92·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s + 1/2·4-s + 0.952·6-s − 0.755·7-s + 0.353·8-s + 0.814·9-s + 0.301·11-s + 0.673·12-s + 1.53·13-s − 0.534·14-s + 1/4·16-s + 0.656·17-s + 0.576·18-s − 0.241·19-s − 1.01·21-s + 0.213·22-s + 0.970·23-s + 0.476·24-s + 1.08·26-s − 0.249·27-s − 0.377·28-s + 0.768·29-s + 0.677·31-s + 0.176·32-s + 0.406·33-s + 0.464·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.091445620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.091445620\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 72 T + p^{3} T^{2} \) |
| 17 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 107 T + p^{3} T^{2} \) |
| 29 | \( 1 - 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 117 T + p^{3} T^{2} \) |
| 37 | \( 1 - 201 T + p^{3} T^{2} \) |
| 41 | \( 1 + 228 T + p^{3} T^{2} \) |
| 43 | \( 1 - 242 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 + 458 T + p^{3} T^{2} \) |
| 59 | \( 1 - 435 T + p^{3} T^{2} \) |
| 61 | \( 1 + 668 T + p^{3} T^{2} \) |
| 67 | \( 1 + 439 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1113 T + p^{3} T^{2} \) |
| 73 | \( 1 - 72 T + p^{3} T^{2} \) |
| 79 | \( 1 + 70 T + p^{3} T^{2} \) |
| 83 | \( 1 + 358 T + p^{3} T^{2} \) |
| 89 | \( 1 - 895 T + p^{3} T^{2} \) |
| 97 | \( 1 + 409 T + p^{3} T^{2} \) |
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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
−10.33540043695310768964816808076, −9.352381424190393499172925001608, −8.618662480590521082880272344921, −7.78062038881583054018405660268, −6.66569123451488391360990598783, −5.87698859401349379582342739083, −4.38808388688918102985928589267, −3.40783959441446585018706007061, −2.83402872586070701409874569682, −1.32783042417446847208695650074,
1.32783042417446847208695650074, 2.83402872586070701409874569682, 3.40783959441446585018706007061, 4.38808388688918102985928589267, 5.87698859401349379582342739083, 6.66569123451488391360990598783, 7.78062038881583054018405660268, 8.618662480590521082880272344921, 9.352381424190393499172925001608, 10.33540043695310768964816808076
