| L(s) = 1 | − 1.43·2-s + 0.0491·4-s + 5-s − 1.25·7-s + 2.79·8-s − 1.43·10-s − 1.47·11-s + 13-s + 1.79·14-s − 4.09·16-s − 3.53·17-s + 4.76·19-s + 0.0491·20-s + 2.10·22-s + 6.94·23-s + 25-s − 1.43·26-s − 0.0614·28-s − 9.61·29-s − 0.103·31-s + 0.278·32-s + 5.06·34-s − 1.25·35-s − 9.80·37-s − 6.81·38-s + 2.79·40-s + 6.15·41-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.0245·4-s + 0.447·5-s − 0.472·7-s + 0.987·8-s − 0.452·10-s − 0.444·11-s + 0.277·13-s + 0.478·14-s − 1.02·16-s − 0.857·17-s + 1.09·19-s + 0.0109·20-s + 0.449·22-s + 1.44·23-s + 0.200·25-s − 0.280·26-s − 0.0116·28-s − 1.78·29-s − 0.0185·31-s + 0.0491·32-s + 0.867·34-s − 0.211·35-s − 1.61·37-s − 1.10·38-s + 0.441·40-s + 0.960·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 + 9.61T + 29T^{2} \) |
| 31 | \( 1 + 0.103T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 - 2.11T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 + 4.08T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 + 4.08T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.954170073639463888802513889505, −7.14094168857779327711308826731, −6.77020707144098311150165551517, −5.54724357206853218216143248513, −5.12709221081699660350323429832, −4.06242869502530944940034983838, −3.16979036056249462816649820389, −2.12511378639634650455573841265, −1.16912409910968007063717144268, 0,
1.16912409910968007063717144268, 2.12511378639634650455573841265, 3.16979036056249462816649820389, 4.06242869502530944940034983838, 5.12709221081699660350323429832, 5.54724357206853218216143248513, 6.77020707144098311150165551517, 7.14094168857779327711308826731, 7.954170073639463888802513889505
