| L(s) = 1 | + 2-s + 2.35·3-s + 4-s + 2.35·6-s + 4.21·7-s + 8-s + 2.56·9-s + 3.59·11-s + 2.35·12-s + 0.585·13-s + 4.21·14-s + 16-s − 4.93·17-s + 2.56·18-s − 5.60·19-s + 9.94·21-s + 3.59·22-s − 6.62·23-s + 2.35·24-s + 0.585·26-s − 1.03·27-s + 4.21·28-s + 29-s − 5.81·31-s + 32-s + 8.48·33-s − 4.93·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.36·3-s + 0.5·4-s + 0.962·6-s + 1.59·7-s + 0.353·8-s + 0.854·9-s + 1.08·11-s + 0.680·12-s + 0.162·13-s + 1.12·14-s + 0.250·16-s − 1.19·17-s + 0.604·18-s − 1.28·19-s + 2.17·21-s + 0.767·22-s − 1.38·23-s + 0.481·24-s + 0.114·26-s − 0.198·27-s + 0.797·28-s + 0.185·29-s − 1.04·31-s + 0.176·32-s + 1.47·33-s − 0.846·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.723769842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.723769842\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 7 | \( 1 - 4.21T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 1.26T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.507T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 - 0.0605T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 7.57T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 0.111T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.181842607370803222796280531311, −8.624982557522753051796454866078, −8.073867170280442892206868683842, −7.16962578358912000591343381874, −6.29824743970719115402974957429, −5.12905422674910408212861569253, −4.15541655367027401525327558735, −3.73307818303695828441508193842, −2.18854444201478814741269132567, −1.82349081547964929363993815727,
1.82349081547964929363993815727, 2.18854444201478814741269132567, 3.73307818303695828441508193842, 4.15541655367027401525327558735, 5.12905422674910408212861569253, 6.29824743970719115402974957429, 7.16962578358912000591343381874, 8.073867170280442892206868683842, 8.624982557522753051796454866078, 9.181842607370803222796280531311
