| L(s) = 1 | + 1.17·2-s − 0.617·4-s − 3.63·5-s − 0.818·7-s − 3.07·8-s − 4.27·10-s − 11-s + 3.69·13-s − 0.962·14-s − 2.38·16-s − 2.16·17-s + 6.52·19-s + 2.24·20-s − 1.17·22-s + 2.86·23-s + 8.24·25-s + 4.34·26-s + 0.505·28-s + 8.99·29-s + 4.22·31-s + 3.35·32-s − 2.54·34-s + 2.97·35-s − 9.05·37-s + 7.67·38-s + 11.1·40-s − 5.62·41-s + ⋯ |
| L(s) = 1 | + 0.831·2-s − 0.308·4-s − 1.62·5-s − 0.309·7-s − 1.08·8-s − 1.35·10-s − 0.301·11-s + 1.02·13-s − 0.257·14-s − 0.596·16-s − 0.524·17-s + 1.49·19-s + 0.502·20-s − 0.250·22-s + 0.597·23-s + 1.64·25-s + 0.851·26-s + 0.0954·28-s + 1.67·29-s + 0.758·31-s + 0.592·32-s − 0.436·34-s + 0.503·35-s − 1.48·37-s + 1.24·38-s + 1.77·40-s − 0.878·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 + 0.818T + 7T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 - 8.99T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 - 6.74T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 2.32T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−7.77308844945664733531769423252, −6.87985852512500885051445496942, −6.34478453877311593670429399756, −5.28616667069068336597102801186, −4.74552815051006429158162346597, −4.05791758969532994664533531908, −3.26950947018102276387675545551, −2.99727601693885689838823606109, −1.10174323465576134481561842489, 0,
1.10174323465576134481561842489, 2.99727601693885689838823606109, 3.26950947018102276387675545551, 4.05791758969532994664533531908, 4.74552815051006429158162346597, 5.28616667069068336597102801186, 6.34478453877311593670429399756, 6.87985852512500885051445496942, 7.77308844945664733531769423252
