| L(s) = 1 | + 0.183·2-s − 1.96·4-s + 0.926·5-s − 0.385·7-s − 0.727·8-s + 0.170·10-s + 0.749·11-s + 2.20·13-s − 0.0707·14-s + 3.79·16-s − 1.34·17-s − 3.59·19-s − 1.82·20-s + 0.137·22-s + 23-s − 4.14·25-s + 0.404·26-s + 0.758·28-s + 29-s − 1.24·31-s + 2.15·32-s − 0.247·34-s − 0.357·35-s + 3.97·37-s − 0.658·38-s − 0.674·40-s − 9.73·41-s + ⋯ |
| L(s) = 1 | + 0.129·2-s − 0.983·4-s + 0.414·5-s − 0.145·7-s − 0.257·8-s + 0.0537·10-s + 0.226·11-s + 0.610·13-s − 0.0189·14-s + 0.949·16-s − 0.326·17-s − 0.823·19-s − 0.407·20-s + 0.0293·22-s + 0.208·23-s − 0.828·25-s + 0.0792·26-s + 0.143·28-s + 0.185·29-s − 0.223·31-s + 0.380·32-s − 0.0423·34-s − 0.0604·35-s + 0.652·37-s − 0.106·38-s − 0.106·40-s − 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.183T + 2T^{2} \) |
| 5 | \( 1 - 0.926T + 5T^{2} \) |
| 7 | \( 1 + 0.385T + 7T^{2} \) |
| 11 | \( 1 - 0.749T + 11T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 + 9.73T + 41T^{2} \) |
| 43 | \( 1 - 4.45T + 43T^{2} \) |
| 47 | \( 1 + 0.400T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 0.0802T + 59T^{2} \) |
| 61 | \( 1 - 3.33T + 61T^{2} \) |
| 67 | \( 1 - 9.55T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 2.45T + 89T^{2} \) |
| 97 | \( 1 + 0.126T + 97T^{2} \) |
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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
−7.965999608863368504716914150195, −6.86385265414216787424016785735, −6.23883579862198311228462999991, −5.55800321115119287963600515801, −4.81258350276708400368945457682, −4.04317187720064267653700045455, −3.43786223486363119221746125838, −2.31952484875978936120011763600, −1.26149258139602070093908203531, 0,
1.26149258139602070093908203531, 2.31952484875978936120011763600, 3.43786223486363119221746125838, 4.04317187720064267653700045455, 4.81258350276708400368945457682, 5.55800321115119287963600515801, 6.23883579862198311228462999991, 6.86385265414216787424016785735, 7.965999608863368504716914150195
