| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·7-s − 2·9-s − 11-s + 2·12-s + 4·14-s − 4·16-s + 2·17-s + 4·18-s − 2·21-s + 2·22-s + 23-s − 5·27-s − 4·28-s − 7·31-s + 8·32-s − 33-s − 4·34-s − 4·36-s + 3·37-s + 8·41-s + 4·42-s + 6·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.577·12-s + 1.06·14-s − 16-s + 0.485·17-s + 0.942·18-s − 0.436·21-s + 0.426·22-s + 0.208·23-s − 0.962·27-s − 0.755·28-s − 1.25·31-s + 1.41·32-s − 0.174·33-s − 0.685·34-s − 2/3·36-s + 0.493·37-s + 1.24·41-s + 0.617·42-s + 0.914·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.90152660010654, −14.34274819351267, −13.97078239187442, −13.24808981691672, −12.87826584065599, −12.26933634160871, −11.52626214162888, −11.06390550171404, −10.58919022415179, −10.01350579011520, −9.429250141756799, −9.190024986194195, −8.642762641785787, −8.108972537669557, −7.561695001327300, −7.197955977734623, −6.480795885041061, −5.751398067788569, −5.340426605418325, −4.249123848732645, −3.760075305582432, −2.790665736150156, −2.563673138561557, −1.641766533995760, −0.7784065168698630, 0,
0.7784065168698630, 1.641766533995760, 2.563673138561557, 2.790665736150156, 3.760075305582432, 4.249123848732645, 5.340426605418325, 5.751398067788569, 6.480795885041061, 7.197955977734623, 7.561695001327300, 8.108972537669557, 8.642762641785787, 9.190024986194195, 9.429250141756799, 10.01350579011520, 10.58919022415179, 11.06390550171404, 11.52626214162888, 12.26933634160871, 12.87826584065599, 13.24808981691672, 13.97078239187442, 14.34274819351267, 14.90152660010654
