| L(s) = 1 | − 2.15·2-s + 0.792·3-s + 2.64·4-s − 1.70·6-s − 4.02·7-s − 1.39·8-s − 2.37·9-s + 1.26·11-s + 2.09·12-s − 4.88·13-s + 8.66·14-s − 2.29·16-s − 0.117·17-s + 5.11·18-s + 6.32·19-s − 3.18·21-s − 2.72·22-s + 2.32·23-s − 1.10·24-s + 10.5·26-s − 4.25·27-s − 10.6·28-s + 3.41·29-s − 5.96·31-s + 7.72·32-s + 1.00·33-s + 0.253·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.457·3-s + 1.32·4-s − 0.697·6-s − 1.51·7-s − 0.492·8-s − 0.790·9-s + 0.381·11-s + 0.605·12-s − 1.35·13-s + 2.31·14-s − 0.572·16-s − 0.0285·17-s + 1.20·18-s + 1.45·19-s − 0.695·21-s − 0.581·22-s + 0.485·23-s − 0.225·24-s + 2.06·26-s − 0.819·27-s − 2.01·28-s + 0.634·29-s − 1.07·31-s + 1.36·32-s + 0.174·33-s + 0.0434·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 - 0.792T + 3T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 + 0.117T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 + 5.96T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 + 2.69T + 73T^{2} \) |
| 79 | \( 1 - 5.79T + 79T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 4.37T + 97T^{2} \) |
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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
−7.68079289262315085252333485938, −7.35826182699661922397990861503, −6.59860180057488417242158223161, −5.86322847464677955529961546112, −4.91203983003497917708976082423, −3.71079055179666577535425643683, −2.86549193967537508868023318702, −2.36982342936359726416108412138, −0.966105713797982667206914148027, 0,
0.966105713797982667206914148027, 2.36982342936359726416108412138, 2.86549193967537508868023318702, 3.71079055179666577535425643683, 4.91203983003497917708976082423, 5.86322847464677955529961546112, 6.59860180057488417242158223161, 7.35826182699661922397990861503, 7.68079289262315085252333485938
