| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 13-s − 14-s + 16-s + 5·17-s + 4·19-s − 20-s + 25-s − 26-s + 28-s + 9·29-s − 32-s − 5·34-s − 35-s − 2·37-s − 4·38-s + 40-s + 2·41-s + 43-s − 4·47-s − 6·49-s − 50-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 + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.67·29-s − 0.176·32-s − 0.857·34-s − 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.312·41-s + 0.152·43-s − 0.583·47-s − 6/7·49-s − 0.141·50-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
| \(L(1)\) |
\(\approx\) |
\(2.486169380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486169380\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.38920059556678, −12.84197452838714, −12.15894160448447, −11.96537385728165, −11.54539201240746, −10.88183506511619, −10.57277466497194, −9.901313153977773, −9.676066373278332, −8.990488478725254, −8.419988291659338, −8.112648005869253, −7.632388268297161, −7.146613002787727, −6.595788080904041, −6.046961520570649, −5.395693034863881, −4.955084393424818, −4.328990013165374, −3.469087170835013, −3.266230316525201, −2.464920185183022, −1.778160628230862, −0.9598149648014495, −0.6761536612482981,
0.6761536612482981, 0.9598149648014495, 1.778160628230862, 2.464920185183022, 3.266230316525201, 3.469087170835013, 4.328990013165374, 4.955084393424818, 5.395693034863881, 6.046961520570649, 6.595788080904041, 7.146613002787727, 7.632388268297161, 8.112648005869253, 8.419988291659338, 8.990488478725254, 9.676066373278332, 9.901313153977773, 10.57277466497194, 10.88183506511619, 11.54539201240746, 11.96537385728165, 12.15894160448447, 12.84197452838714, 13.38920059556678
