| L(s) = 1 | − 1.61·2-s + 0.618·4-s − 5-s − 4.23·7-s + 2.23·8-s + 1.61·10-s − 0.236·11-s + 13-s + 6.85·14-s − 4.85·16-s + 7.47·17-s + 2.47·19-s − 0.618·20-s + 0.381·22-s + 4.47·23-s + 25-s − 1.61·26-s − 2.61·28-s − 29-s − 8·31-s + 3.38·32-s − 12.0·34-s + 4.23·35-s − 4.00·38-s − 2.23·40-s − 6·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s − 0.447·5-s − 1.60·7-s + 0.790·8-s + 0.511·10-s − 0.0711·11-s + 0.277·13-s + 1.83·14-s − 1.21·16-s + 1.81·17-s + 0.567·19-s − 0.138·20-s + 0.0814·22-s + 0.932·23-s + 0.200·25-s − 0.317·26-s − 0.494·28-s − 0.185·29-s − 1.43·31-s + 0.597·32-s − 2.07·34-s + 0.716·35-s − 0.648·38-s − 0.353·40-s − 0.937·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 3.76T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 8.47T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 + 9.94T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−9.239748993847396031045299579043, −8.653453618454288766397376057688, −7.55075159942762812595657162738, −7.19652304162427331253874108891, −6.08738029440758045555188525027, −5.10215716093478256675803103231, −3.71842405262805641037485466416, −3.05785282940764957868027393824, −1.28568299950952581982165489920, 0,
1.28568299950952581982165489920, 3.05785282940764957868027393824, 3.71842405262805641037485466416, 5.10215716093478256675803103231, 6.08738029440758045555188525027, 7.19652304162427331253874108891, 7.55075159942762812595657162738, 8.653453618454288766397376057688, 9.239748993847396031045299579043
