| L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s + 13-s − 2·14-s + 16-s + 8·17-s + 6·19-s + 20-s − 6·23-s + 25-s + 26-s − 2·28-s − 4·29-s + 32-s + 8·34-s − 2·35-s − 2·37-s + 6·38-s + 40-s − 2·41-s + 4·43-s − 6·46-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.94·17-s + 1.37·19-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.742·29-s + 0.176·32-s + 1.37·34-s − 0.338·35-s − 0.328·37-s + 0.973·38-s + 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.884·46-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.143333006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.143333006\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.55511115902300, −12.93674192706077, −12.40559624187784, −12.08907731178501, −11.70646225495494, −11.03539462227748, −10.50376525778672, −9.975515031493470, −9.589086629188112, −9.356238486799610, −8.377229244934904, −7.939730311127344, −7.510027054203594, −6.831223417931529, −6.472277156123206, −5.743215971696785, −5.419936085482400, −5.189769435066361, −4.096518050656935, −3.714503654594025, −3.272463434651813, −2.683369434829506, −1.989259411775321, −1.271804887625846, −0.6219828876461286,
0.6219828876461286, 1.271804887625846, 1.989259411775321, 2.683369434829506, 3.272463434651813, 3.714503654594025, 4.096518050656935, 5.189769435066361, 5.419936085482400, 5.743215971696785, 6.472277156123206, 6.831223417931529, 7.510027054203594, 7.939730311127344, 8.377229244934904, 9.356238486799610, 9.589086629188112, 9.975515031493470, 10.50376525778672, 11.03539462227748, 11.70646225495494, 12.08907731178501, 12.40559624187784, 12.93674192706077, 13.55511115902300
