| L(s) = 1 | − 0.599·2-s − 1.64·4-s − 2.41·5-s + 4.19·7-s + 2.18·8-s + 1.45·10-s − 1.92·11-s + 4.86·13-s − 2.51·14-s + 1.97·16-s + 1.56·19-s + 3.96·20-s + 1.15·22-s − 7.71·23-s + 0.848·25-s − 2.91·26-s − 6.88·28-s + 7.69·29-s − 9.20·31-s − 5.54·32-s − 10.1·35-s + 3.98·37-s − 0.941·38-s − 5.28·40-s − 10.2·41-s + 6.58·43-s + 3.16·44-s + ⋯ |
| L(s) = 1 | − 0.424·2-s − 0.820·4-s − 1.08·5-s + 1.58·7-s + 0.772·8-s + 0.458·10-s − 0.581·11-s + 1.34·13-s − 0.673·14-s + 0.492·16-s + 0.360·19-s + 0.886·20-s + 0.246·22-s − 1.60·23-s + 0.169·25-s − 0.572·26-s − 1.30·28-s + 1.42·29-s − 1.65·31-s − 0.980·32-s − 1.71·35-s + 0.654·37-s − 0.152·38-s − 0.834·40-s − 1.59·41-s + 1.00·43-s + 0.477·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.599T + 2T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 - 4.86T + 13T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 - 7.69T + 29T^{2} \) |
| 31 | \( 1 + 9.20T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + 9.34T + 47T^{2} \) |
| 53 | \( 1 + 0.192T + 53T^{2} \) |
| 59 | \( 1 + 0.0869T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 + 1.95T + 67T^{2} \) |
| 71 | \( 1 - 5.73T + 71T^{2} \) |
| 73 | \( 1 - 5.81T + 73T^{2} \) |
| 79 | \( 1 + 4.18T + 79T^{2} \) |
| 83 | \( 1 - 4.64T + 83T^{2} \) |
| 89 | \( 1 - 4.84T + 89T^{2} \) |
| 97 | \( 1 + 2.50T + 97T^{2} \) |
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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
−7.977452835884772722634058242177, −7.14377624431815528748572599328, −6.00935548293157923920759212188, −5.25705188077253918610071251846, −4.59603199993147256880480295897, −4.02052200846216887751844788221, −3.37297153588564886426989949921, −1.93307545466236342856082615927, −1.14418947924543780826353171252, 0,
1.14418947924543780826353171252, 1.93307545466236342856082615927, 3.37297153588564886426989949921, 4.02052200846216887751844788221, 4.59603199993147256880480295897, 5.25705188077253918610071251846, 6.00935548293157923920759212188, 7.14377624431815528748572599328, 7.977452835884772722634058242177
