| L(s) = 1 | + 1.34·2-s − 2.29·3-s − 0.190·4-s − 3.94·5-s − 3.08·6-s − 5.02·7-s − 2.94·8-s + 2.27·9-s − 5.31·10-s − 11-s + 0.436·12-s − 2.87·13-s − 6.75·14-s + 9.06·15-s − 3.58·16-s − 17-s + 3.06·18-s + 6.17·19-s + 0.750·20-s + 11.5·21-s − 1.34·22-s − 4.25·23-s + 6.76·24-s + 10.5·25-s − 3.86·26-s + 1.66·27-s + 0.954·28-s + ⋯ |
| L(s) = 1 | + 0.951·2-s − 1.32·3-s − 0.0950·4-s − 1.76·5-s − 1.26·6-s − 1.89·7-s − 1.04·8-s + 0.758·9-s − 1.67·10-s − 0.301·11-s + 0.126·12-s − 0.797·13-s − 1.80·14-s + 2.34·15-s − 0.895·16-s − 0.242·17-s + 0.721·18-s + 1.41·19-s + 0.167·20-s + 2.51·21-s − 0.286·22-s − 0.886·23-s + 1.38·24-s + 2.11·25-s − 0.758·26-s + 0.320·27-s + 0.180·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 + 5.02T + 7T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 19 | \( 1 - 6.17T + 19T^{2} \) |
| 23 | \( 1 + 4.25T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 + 6.98T + 41T^{2} \) |
| 47 | \( 1 + 0.315T + 47T^{2} \) |
| 53 | \( 1 + 9.81T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 9.17T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.39848029190125385104886060661, −6.38830046353099631472854595754, −6.22379176154723931433862620651, −5.18052654805796777088838217621, −4.80859528239009715154148450686, −3.88290829679228798780242812229, −3.42921240969012180830811509006, −2.77572202595021638131810588841, −0.55915023702349573121672492896, 0,
0.55915023702349573121672492896, 2.77572202595021638131810588841, 3.42921240969012180830811509006, 3.88290829679228798780242812229, 4.80859528239009715154148450686, 5.18052654805796777088838217621, 6.22379176154723931433862620651, 6.38830046353099631472854595754, 7.39848029190125385104886060661
