| L(s) = 1 | − 2.41·2-s − 1.38·3-s + 3.84·4-s + 1.63·5-s + 3.34·6-s − 3.98·7-s − 4.45·8-s − 1.08·9-s − 3.95·10-s + 3.77·11-s − 5.31·12-s − 0.915·13-s + 9.62·14-s − 2.26·15-s + 3.08·16-s + 2.97·17-s + 2.62·18-s + 5.78·19-s + 6.28·20-s + 5.51·21-s − 9.11·22-s − 4.09·23-s + 6.16·24-s − 2.32·25-s + 2.21·26-s + 5.65·27-s − 15.3·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.799·3-s + 1.92·4-s + 0.731·5-s + 1.36·6-s − 1.50·7-s − 1.57·8-s − 0.361·9-s − 1.24·10-s + 1.13·11-s − 1.53·12-s − 0.253·13-s + 2.57·14-s − 0.584·15-s + 0.770·16-s + 0.721·17-s + 0.617·18-s + 1.32·19-s + 1.40·20-s + 1.20·21-s − 1.94·22-s − 0.853·23-s + 1.25·24-s − 0.465·25-s + 0.433·26-s + 1.08·27-s − 2.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 + 0.915T + 13T^{2} \) |
| 17 | \( 1 - 2.97T + 17T^{2} \) |
| 19 | \( 1 - 5.78T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 - 8.51T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 - 2.34T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 - 1.34T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + 5.63T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 - 7.86T + 97T^{2} \) |
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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
−7.57837919624527017834540095804, −6.59733373908719316387329038689, −6.44247929516060718999340451646, −5.84088063481966181025324256856, −4.94865707122504397960862490898, −3.54650851621038219891731776558, −2.91374154267485414888798398679, −1.81778513273596692368297359538, −0.935757237890074650988008140624, 0,
0.935757237890074650988008140624, 1.81778513273596692368297359538, 2.91374154267485414888798398679, 3.54650851621038219891731776558, 4.94865707122504397960862490898, 5.84088063481966181025324256856, 6.44247929516060718999340451646, 6.59733373908719316387329038689, 7.57837919624527017834540095804
