| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 4·11-s − 6·13-s − 14-s + 16-s + 2·17-s + 19-s + 20-s − 4·22-s + 25-s + 6·26-s + 28-s − 2·29-s − 8·31-s − 32-s − 2·34-s + 35-s − 2·37-s − 38-s − 40-s + 10·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.229·19-s + 0.223·20-s − 0.852·22-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.169·35-s − 0.328·37-s − 0.162·38-s − 0.158·40-s + 1.56·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.82315617683786, −16.36543187085917, −15.55294803028208, −14.82797305749471, −14.42511768492975, −14.17938398591376, −13.13554052410183, −12.54515288634340, −12.00659746702208, −11.48507370424141, −10.86247209139284, −10.15685148662319, −9.535050528622214, −9.294170644819279, −8.556505706768082, −7.737415690735281, −7.270982980395965, −6.709015155124581, −5.877073171886998, −5.270885704548557, −4.505623940433223, −3.631572233417939, −2.747951110112853, −1.921944561891771, −1.259809905071680, 0,
1.259809905071680, 1.921944561891771, 2.747951110112853, 3.631572233417939, 4.505623940433223, 5.270885704548557, 5.877073171886998, 6.709015155124581, 7.270982980395965, 7.737415690735281, 8.556505706768082, 9.294170644819279, 9.535050528622214, 10.15685148662319, 10.86247209139284, 11.48507370424141, 12.00659746702208, 12.54515288634340, 13.13554052410183, 14.17938398591376, 14.42511768492975, 14.82797305749471, 15.55294803028208, 16.36543187085917, 16.82315617683786
