| L(s) = 1 | + 2-s − 2.47·3-s + 4-s − 0.567·5-s − 2.47·6-s − 4.82·7-s + 8-s + 3.13·9-s − 0.567·10-s + 1.83·11-s − 2.47·12-s − 5.69·13-s − 4.82·14-s + 1.40·15-s + 16-s + 4.15·17-s + 3.13·18-s + 19-s − 0.567·20-s + 11.9·21-s + 1.83·22-s + 3.32·23-s − 2.47·24-s − 4.67·25-s − 5.69·26-s − 0.333·27-s − 4.82·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s − 0.253·5-s − 1.01·6-s − 1.82·7-s + 0.353·8-s + 1.04·9-s − 0.179·10-s + 0.552·11-s − 0.714·12-s − 1.58·13-s − 1.28·14-s + 0.362·15-s + 0.250·16-s + 1.00·17-s + 0.738·18-s + 0.229·19-s − 0.126·20-s + 2.60·21-s + 0.390·22-s + 0.693·23-s − 0.505·24-s − 0.935·25-s − 1.11·26-s − 0.0641·27-s − 0.911·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 + 0.567T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 - 0.666T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + 1.66T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 0.920T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 + 6.95T + 53T^{2} \) |
| 59 | \( 1 + 9.75T + 59T^{2} \) |
| 61 | \( 1 - 3.51T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 0.562T + 73T^{2} \) |
| 79 | \( 1 - 7.21T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 - 6.85T + 97T^{2} \) |
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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
−7.29598724657043021987111758467, −6.47734861547441512344195495295, −6.15039859091400527183776548689, −5.44916939344132358396237643653, −4.81994362431783415170646619748, −3.98832653693840574849692369065, −3.24098293013627034149577299388, −2.47729176210984530926167934155, −0.959177492867094051376393720103, 0,
0.959177492867094051376393720103, 2.47729176210984530926167934155, 3.24098293013627034149577299388, 3.98832653693840574849692369065, 4.81994362431783415170646619748, 5.44916939344132358396237643653, 6.15039859091400527183776548689, 6.47734861547441512344195495295, 7.29598724657043021987111758467
