| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 3·9-s − 10-s + 6·13-s + 4·14-s + 16-s − 6·17-s − 3·18-s + 2·19-s − 20-s − 8·23-s + 25-s + 6·26-s + 4·28-s + 29-s + 4·31-s + 32-s − 6·34-s − 4·35-s − 3·36-s − 12·37-s + 2·38-s − 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 9-s − 0.316·10-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.458·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.755·28-s + 0.185·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.676·35-s − 1/2·36-s − 1.97·37-s + 0.324·38-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.982615417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.982615417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.60448187531757, −14.06655246656280, −13.92986367918383, −13.47660777496398, −12.64719708667170, −11.99724666500373, −11.70293031984370, −11.18066603548434, −10.78205434609587, −10.46052092306947, −9.291742887691760, −8.690302785469336, −8.412507008659434, −7.829289566285313, −7.338074420966369, −6.304507046206853, −6.170210701233944, −5.356090819281905, −4.875408631188005, −4.081214674221239, −3.871738684321786, −2.943014246910961, −2.187431362079829, −1.608667702917205, −0.6447513564971381,
0.6447513564971381, 1.608667702917205, 2.187431362079829, 2.943014246910961, 3.871738684321786, 4.081214674221239, 4.875408631188005, 5.356090819281905, 6.170210701233944, 6.304507046206853, 7.338074420966369, 7.829289566285313, 8.412507008659434, 8.690302785469336, 9.291742887691760, 10.46052092306947, 10.78205434609587, 11.18066603548434, 11.70293031984370, 11.99724666500373, 12.64719708667170, 13.47660777496398, 13.92986367918383, 14.06655246656280, 14.60448187531757
