L(s) = 1 | − 2-s + 2.77·3-s + 4-s + 0.377·5-s − 2.77·6-s + 0.478·7-s − 8-s + 4.67·9-s − 0.377·10-s − 4.92·11-s + 2.77·12-s − 1.38·13-s − 0.478·14-s + 1.04·15-s + 16-s − 5.72·17-s − 4.67·18-s − 19-s + 0.377·20-s + 1.32·21-s + 4.92·22-s + 7.43·23-s − 2.77·24-s − 4.85·25-s + 1.38·26-s + 4.63·27-s + 0.478·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.59·3-s + 0.5·4-s + 0.169·5-s − 1.13·6-s + 0.180·7-s − 0.353·8-s + 1.55·9-s − 0.119·10-s − 1.48·11-s + 0.799·12-s − 0.383·13-s − 0.127·14-s + 0.270·15-s + 0.250·16-s − 1.38·17-s − 1.10·18-s − 0.229·19-s + 0.0845·20-s + 0.289·21-s + 1.05·22-s + 1.55·23-s − 0.565·24-s − 0.971·25-s + 0.271·26-s + 0.892·27-s + 0.0903·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 - 0.377T + 5T^{2} \) |
| 7 | \( 1 - 0.478T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 + 0.491T + 31T^{2} \) |
| 37 | \( 1 - 0.0388T + 37T^{2} \) |
| 41 | \( 1 + 0.959T + 41T^{2} \) |
| 43 | \( 1 + 6.69T + 43T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 - 0.976T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 0.324T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.85774120811771810983681177659, −7.03212229572036243252842727818, −6.46886125320221581238284038629, −5.25055976235591222684814820899, −4.64734196400537702617720817259, −3.61643462418144777952568976144, −2.71672762212899329116987233617, −2.42098965008141248572360697896, −1.52054441727227531419663054180, 0,
1.52054441727227531419663054180, 2.42098965008141248572360697896, 2.71672762212899329116987233617, 3.61643462418144777952568976144, 4.64734196400537702617720817259, 5.25055976235591222684814820899, 6.46886125320221581238284038629, 7.03212229572036243252842727818, 7.85774120811771810983681177659