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
−9.095932902205181179578372046337, −7.993753500361472324876948423776, −7.56872729140441273204203570227, −6.48479365108209866322355732707, −5.54119792924237596299815788548, −5.09749030201889466224237422497, −3.72978536930896817466808366790, −2.97257568923183612670383180011, −2.05629181597688770640909017754, −0.991680808067481084250114185412,
0.62674347670875099260631967967, 2.25220788030208594698013464688, 3.70871785708933881972579842798, 4.08138226033058356006880172201, 5.14965293789467323514001022207, 5.73925305197396526310405932956, 6.84540714943143656177757857363, 7.26876052801828573588868265475, 8.193278888294562177613671787076, 8.972896758696052883051112482108