| L(s) = 1 | − 1.65·5-s − 2.41·7-s + 5.94·11-s − 3.22·13-s − 3·17-s + 6.63·19-s − 2.94·23-s − 2.26·25-s + 1.29·29-s + 0.588·31-s + 3.98·35-s + 0.0418·37-s + 4.90·41-s + 5.18·43-s − 3.73·47-s − 1.18·49-s − 11.6·53-s − 9.82·55-s − 7.34·59-s + 11.0·61-s + 5.33·65-s − 1.85·67-s − 5.51·71-s + 5.55·73-s − 14.3·77-s + 3.78·79-s + 3.98·83-s + ⋯ |
| L(s) = 1 | − 0.739·5-s − 0.911·7-s + 1.79·11-s − 0.894·13-s − 0.727·17-s + 1.52·19-s − 0.613·23-s − 0.453·25-s + 0.239·29-s + 0.105·31-s + 0.673·35-s + 0.00688·37-s + 0.765·41-s + 0.790·43-s − 0.545·47-s − 0.169·49-s − 1.59·53-s − 1.32·55-s − 0.956·59-s + 1.41·61-s + 0.661·65-s − 0.226·67-s − 0.654·71-s + 0.650·73-s − 1.63·77-s + 0.425·79-s + 0.437·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 5.94T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 - 0.588T + 31T^{2} \) |
| 37 | \( 1 - 0.0418T + 37T^{2} \) |
| 41 | \( 1 - 4.90T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 - 3.78T + 79T^{2} \) |
| 83 | \( 1 - 3.98T + 83T^{2} \) |
| 89 | \( 1 + 8.15T + 89T^{2} \) |
| 97 | \( 1 - 0.260T + 97T^{2} \) |
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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
−7.986510239068404212924453820183, −7.36901086495246683706311938885, −6.62881203407608945484571557486, −6.10289652796523199982805761756, −4.99473379624509408119723929810, −4.10319146589635572577495966231, −3.57230502601571549533672351001, −2.62758414343559060131625559676, −1.30793952740054271621150881334, 0,
1.30793952740054271621150881334, 2.62758414343559060131625559676, 3.57230502601571549533672351001, 4.10319146589635572577495966231, 4.99473379624509408119723929810, 6.10289652796523199982805761756, 6.62881203407608945484571557486, 7.36901086495246683706311938885, 7.986510239068404212924453820183
