| L(s) = 1 | + 0.703·2-s − 1.50·4-s − 1.68·5-s − 3.24·7-s − 2.46·8-s − 1.18·10-s + 2.04·11-s + 0.236·13-s − 2.28·14-s + 1.27·16-s + 3.00·17-s + 4.79·19-s + 2.53·20-s + 1.43·22-s − 2.15·25-s + 0.166·26-s + 4.88·28-s + 7.10·29-s + 6.70·31-s + 5.82·32-s + 2.11·34-s + 5.47·35-s − 11.4·37-s + 3.37·38-s + 4.15·40-s − 8.20·41-s − 2.88·43-s + ⋯ |
| L(s) = 1 | + 0.497·2-s − 0.752·4-s − 0.754·5-s − 1.22·7-s − 0.871·8-s − 0.375·10-s + 0.616·11-s + 0.0656·13-s − 0.611·14-s + 0.318·16-s + 0.728·17-s + 1.09·19-s + 0.567·20-s + 0.306·22-s − 0.431·25-s + 0.0326·26-s + 0.923·28-s + 1.32·29-s + 1.20·31-s + 1.03·32-s + 0.362·34-s + 0.926·35-s − 1.88·37-s + 0.547·38-s + 0.657·40-s − 1.28·41-s − 0.439·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.703T + 2T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 29 | \( 1 - 7.10T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 8.20T + 41T^{2} \) |
| 43 | \( 1 + 2.88T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 0.531T + 53T^{2} \) |
| 59 | \( 1 - 3.44T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 + 2.09T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 - 2.50T + 89T^{2} \) |
| 97 | \( 1 + 1.81T + 97T^{2} \) |
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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
−7.955739134479940399851023668745, −7.11973131396922218982071962191, −6.41142845066334029284785482591, −5.68730956452124958028212277857, −4.89204174306585648838591069481, −4.05750336198779311586629048244, −3.41455768346285521507069232021, −2.93230627744423243357148879066, −1.12914987083869337799353870533, 0,
1.12914987083869337799353870533, 2.93230627744423243357148879066, 3.41455768346285521507069232021, 4.05750336198779311586629048244, 4.89204174306585648838591069481, 5.68730956452124958028212277857, 6.41142845066334029284785482591, 7.11973131396922218982071962191, 7.955739134479940399851023668745
