| L(s) = 1 | − 2-s + 1.30·3-s + 4-s + 5-s − 1.30·6-s + 5.09·7-s − 8-s − 1.30·9-s − 10-s + 3.73·11-s + 1.30·12-s − 1.53·13-s − 5.09·14-s + 1.30·15-s + 16-s − 6.50·17-s + 1.30·18-s + 7.67·19-s + 20-s + 6.63·21-s − 3.73·22-s − 6.65·23-s − 1.30·24-s + 25-s + 1.53·26-s − 5.60·27-s + 5.09·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.751·3-s + 0.5·4-s + 0.447·5-s − 0.531·6-s + 1.92·7-s − 0.353·8-s − 0.435·9-s − 0.316·10-s + 1.12·11-s + 0.375·12-s − 0.425·13-s − 1.36·14-s + 0.336·15-s + 0.250·16-s − 1.57·17-s + 0.307·18-s + 1.76·19-s + 0.223·20-s + 1.44·21-s − 0.797·22-s − 1.38·23-s − 0.265·24-s + 0.200·25-s + 0.300·26-s − 1.07·27-s + 0.963·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.546190029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546190029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 - 5.09T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 - 7.67T + 19T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 + 0.491T + 31T^{2} \) |
| 37 | \( 1 - 0.176T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 8.28T + 43T^{2} \) |
| 47 | \( 1 - 9.99T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 6.34T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 + 6.77T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 - 6.72T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.574073802025278254284382159007, −7.70827027418327219644620191456, −7.45610055147962946908297367365, −6.29044216062836881554527106799, −5.54846969032724659629702788130, −4.62176480131925065291345880434, −3.83499980489256297858357246043, −2.51104214001161411255270645872, −2.00721259877674964180865162431, −1.04448704838976984587362053117,
1.04448704838976984587362053117, 2.00721259877674964180865162431, 2.51104214001161411255270645872, 3.83499980489256297858357246043, 4.62176480131925065291345880434, 5.54846969032724659629702788130, 6.29044216062836881554527106799, 7.45610055147962946908297367365, 7.70827027418327219644620191456, 8.574073802025278254284382159007
