| L(s) = 1 | + 1.95·2-s − 0.350·3-s + 1.83·4-s − 0.686·6-s − 1.14·7-s − 0.318·8-s − 2.87·9-s + 0.395·11-s − 0.643·12-s + 0.595·13-s − 2.25·14-s − 4.29·16-s + 7.79·17-s − 5.63·18-s + 3.25·19-s + 0.402·21-s + 0.774·22-s − 3.49·23-s + 0.111·24-s + 1.16·26-s + 2.05·27-s − 2.11·28-s + 6.23·29-s − 9.01·31-s − 7.78·32-s − 0.138·33-s + 15.2·34-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 0.202·3-s + 0.918·4-s − 0.280·6-s − 0.434·7-s − 0.112·8-s − 0.959·9-s + 0.119·11-s − 0.185·12-s + 0.165·13-s − 0.601·14-s − 1.07·16-s + 1.88·17-s − 1.32·18-s + 0.746·19-s + 0.0878·21-s + 0.165·22-s − 0.729·23-s + 0.0228·24-s + 0.228·26-s + 0.396·27-s − 0.398·28-s + 1.15·29-s − 1.61·31-s − 1.37·32-s − 0.0241·33-s + 2.61·34-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 | \( 1 - 1.95T + 2T^{2} \) |
| 3 | \( 1 + 0.350T + 3T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 0.395T + 11T^{2} \) |
| 13 | \( 1 - 0.595T + 13T^{2} \) |
| 17 | \( 1 - 7.79T + 17T^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 + 0.667T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 + 9.26T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 + 0.719T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 2.84T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 5.86T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 0.175T + 97T^{2} \) |
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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
−7.64758194607361777013130089459, −6.61315020128576121908675916359, −6.13709772062889196540597577303, −5.39802705117433751938458231232, −5.07051559012859070602649752966, −3.96810248620635034694480086709, −3.24862713424372051958581320685, −2.89669103455973888185804578746, −1.51594561288429304460524357861, 0,
1.51594561288429304460524357861, 2.89669103455973888185804578746, 3.24862713424372051958581320685, 3.96810248620635034694480086709, 5.07051559012859070602649752966, 5.39802705117433751938458231232, 6.13709772062889196540597577303, 6.61315020128576121908675916359, 7.64758194607361777013130089459
