| L(s) = 1 | − 2.60·2-s − 3.29·3-s + 4.78·4-s + 5-s + 8.59·6-s − 7.26·8-s + 7.87·9-s − 2.60·10-s − 3.29·11-s − 15.7·12-s − 4.44·13-s − 3.29·15-s + 9.34·16-s − 5.60·17-s − 20.5·18-s − 3.20·19-s + 4.78·20-s + 8.58·22-s + 0.660·23-s + 23.9·24-s + 25-s + 11.5·26-s − 16.0·27-s − 3.12·29-s + 8.59·30-s − 31-s − 9.82·32-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 1.90·3-s + 2.39·4-s + 0.447·5-s + 3.50·6-s − 2.56·8-s + 2.62·9-s − 0.823·10-s − 0.993·11-s − 4.55·12-s − 1.23·13-s − 0.851·15-s + 2.33·16-s − 1.35·17-s − 4.83·18-s − 0.735·19-s + 1.07·20-s + 1.83·22-s + 0.137·23-s + 4.88·24-s + 0.200·25-s + 2.27·26-s − 3.09·27-s − 0.579·29-s + 1.56·30-s − 0.179·31-s − 1.73·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 3.29T + 3T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 23 | \( 1 - 0.660T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 + 3.15T + 41T^{2} \) |
| 43 | \( 1 + 4.88T + 43T^{2} \) |
| 47 | \( 1 - 7.94T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 0.311T + 61T^{2} \) |
| 67 | \( 1 - 5.01T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 2.43T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 1.28T + 83T^{2} \) |
| 89 | \( 1 - 9.09T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.41501137019959528106291791906, −6.78706047361763105110087280103, −6.57816948062122397994319697482, −5.48364812193952769006584307764, −5.17827629018058971920890316567, −4.10467799199955919385064421415, −2.39821078094285686633650988946, −1.97326190144921362187362160489, −0.70827298417474601205728997125, 0,
0.70827298417474601205728997125, 1.97326190144921362187362160489, 2.39821078094285686633650988946, 4.10467799199955919385064421415, 5.17827629018058971920890316567, 5.48364812193952769006584307764, 6.57816948062122397994319697482, 6.78706047361763105110087280103, 7.41501137019959528106291791906
