| L(s) = 1 | − 2-s − 0.179·3-s + 4-s + 0.789·5-s + 0.179·6-s − 0.160·7-s − 8-s − 2.96·9-s − 0.789·10-s + 4.46·11-s − 0.179·12-s + 6.52·13-s + 0.160·14-s − 0.141·15-s + 16-s + 5.20·17-s + 2.96·18-s + 4.25·19-s + 0.789·20-s + 0.0287·21-s − 4.46·22-s − 23-s + 0.179·24-s − 4.37·25-s − 6.52·26-s + 1.06·27-s − 0.160·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.103·3-s + 0.5·4-s + 0.353·5-s + 0.0731·6-s − 0.0607·7-s − 0.353·8-s − 0.989·9-s − 0.249·10-s + 1.34·11-s − 0.0517·12-s + 1.80·13-s + 0.0429·14-s − 0.0365·15-s + 0.250·16-s + 1.26·17-s + 0.699·18-s + 0.977·19-s + 0.176·20-s + 0.00628·21-s − 0.951·22-s − 0.208·23-s + 0.0365·24-s − 0.875·25-s − 1.27·26-s + 0.205·27-s − 0.0303·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.878188516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878188516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.179T + 3T^{2} \) |
| 5 | \( 1 - 0.789T + 5T^{2} \) |
| 7 | \( 1 + 0.160T + 7T^{2} \) |
| 11 | \( 1 - 4.46T + 11T^{2} \) |
| 13 | \( 1 - 6.52T + 13T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 29 | \( 1 - 9.71T + 29T^{2} \) |
| 31 | \( 1 + 0.361T + 31T^{2} \) |
| 37 | \( 1 - 4.58T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 + 2.15T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 4.44T + 89T^{2} \) |
| 97 | \( 1 + 8.27T + 97T^{2} \) |
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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
−8.173701608549253553959286566008, −7.52541890166214517322670819875, −6.54829957538851471279093123529, −5.94678221325209662670506562865, −5.67292876993177604957636734930, −4.31262966189236268370194886614, −3.46104062398876274885076836828, −2.81550382515513215356750390812, −1.46881735050010987932049928129, −0.918505568588920841429284812750,
0.918505568588920841429284812750, 1.46881735050010987932049928129, 2.81550382515513215356750390812, 3.46104062398876274885076836828, 4.31262966189236268370194886614, 5.67292876993177604957636734930, 5.94678221325209662670506562865, 6.54829957538851471279093123529, 7.52541890166214517322670819875, 8.173701608549253553959286566008
