| L(s) = 1 | − 2-s + 1.41·3-s + 4-s + 0.157·5-s − 1.41·6-s − 8-s − 0.999·9-s − 0.157·10-s + 3.17·11-s + 1.41·12-s − 0.157·13-s + 0.222·15-s + 16-s + 3.23·17-s + 0.999·18-s − 6.22·19-s + 0.157·20-s − 3.17·22-s − 9.39·23-s − 1.41·24-s − 4.97·25-s + 0.157·26-s − 5.65·27-s − 29-s − 0.222·30-s − 9.45·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.816·3-s + 0.5·4-s + 0.0703·5-s − 0.577·6-s − 0.353·8-s − 0.333·9-s − 0.0497·10-s + 0.957·11-s + 0.408·12-s − 0.0436·13-s + 0.0574·15-s + 0.250·16-s + 0.784·17-s + 0.235·18-s − 1.42·19-s + 0.0351·20-s − 0.677·22-s − 1.95·23-s − 0.288·24-s − 0.995·25-s + 0.0308·26-s − 1.08·27-s − 0.185·29-s − 0.0406·30-s − 1.69·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 0.157T + 5T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 0.157T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 23 | \( 1 + 9.39T + 23T^{2} \) |
| 31 | \( 1 + 9.45T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 + 4.39T + 41T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 + 1.97T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 6.57T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 0.942T + 89T^{2} \) |
| 97 | \( 1 + 2.42T + 97T^{2} \) |
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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
−8.607392087170223486098340768557, −7.77628511269386017602520911102, −7.19250833502611168375510794999, −6.09969536628318655783136811850, −5.65071907706083682886583824130, −4.05439562917161984618557032450, −3.61316680534941185293643352845, −2.33260983902299690087688273628, −1.71952934886559228843554518499, 0,
1.71952934886559228843554518499, 2.33260983902299690087688273628, 3.61316680534941185293643352845, 4.05439562917161984618557032450, 5.65071907706083682886583824130, 6.09969536628318655783136811850, 7.19250833502611168375510794999, 7.77628511269386017602520911102, 8.607392087170223486098340768557
