| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 2i·7-s − i·8-s − 9-s − i·12-s + 2i·13-s + 2·14-s + 16-s + i·17-s − i·18-s + 4·19-s + 2·21-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.554i·13-s + 0.534·14-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s + 0.917·19-s + 0.436·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510477192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510477192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.972896758696052883051112482108, −8.193278888294562177613671787076, −7.26876052801828573588868265475, −6.84540714943143656177757857363, −5.73925305197396526310405932956, −5.14965293789467323514001022207, −4.08138226033058356006880172201, −3.70871785708933881972579842798, −2.25220788030208594698013464688, −0.62674347670875099260631967967,
0.991680808067481084250114185412, 2.05629181597688770640909017754, 2.97257568923183612670383180011, 3.72978536930896817466808366790, 5.09749030201889466224237422497, 5.54119792924237596299815788548, 6.48479365108209866322355732707, 7.56872729140441273204203570227, 7.993753500361472324876948423776, 9.095932902205181179578372046337
