| L(s) = 1 | + 2-s + 4-s − 1.46·5-s − 3.53·7-s + 8-s − 1.46·10-s + 4.22·11-s − 0.227·13-s − 3.53·14-s + 16-s − 6.27·17-s + 2.57·19-s − 1.46·20-s + 4.22·22-s − 3.30·23-s − 2.85·25-s − 0.227·26-s − 3.53·28-s + 3.57·29-s + 7.37·31-s + 32-s − 6.27·34-s + 5.18·35-s − 4.39·37-s + 2.57·38-s − 1.46·40-s + 4.53·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.655·5-s − 1.33·7-s + 0.353·8-s − 0.463·10-s + 1.27·11-s − 0.0631·13-s − 0.946·14-s + 0.250·16-s − 1.52·17-s + 0.590·19-s − 0.327·20-s + 0.901·22-s − 0.688·23-s − 0.570·25-s − 0.0446·26-s − 0.668·28-s + 0.664·29-s + 1.32·31-s + 0.176·32-s − 1.07·34-s + 0.876·35-s − 0.723·37-s + 0.417·38-s − 0.231·40-s + 0.708·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.111446067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.111446067\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 + 0.227T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 1.84T + 59T^{2} \) |
| 61 | \( 1 + 0.0848T + 61T^{2} \) |
| 67 | \( 1 - 4.55T + 67T^{2} \) |
| 71 | \( 1 + 2.24T + 71T^{2} \) |
| 73 | \( 1 + 6.79T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 + 1.74T + 83T^{2} \) |
| 89 | \( 1 - 5.71T + 89T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
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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
−7.63707544242179467103677426777, −6.88326924315995065676832977569, −6.44584981124511887516669060240, −5.96550403104940977530930968889, −4.83648885931733921321284439567, −4.15814881720262715290428240449, −3.64195757046924683267122827938, −2.91538363392109785038840933016, −1.96625480050564142542837112356, −0.62590415231152585427114275687,
0.62590415231152585427114275687, 1.96625480050564142542837112356, 2.91538363392109785038840933016, 3.64195757046924683267122827938, 4.15814881720262715290428240449, 4.83648885931733921321284439567, 5.96550403104940977530930968889, 6.44584981124511887516669060240, 6.88326924315995065676832977569, 7.63707544242179467103677426777
