| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 2i·7-s − i·8-s − 9-s + i·12-s − i·13-s − 2·14-s + 16-s − i·18-s − 2·19-s + 2·21-s + 6i·23-s − 24-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 − 0.277i·13-s − 0.534·14-s + 0.250·16-s − 0.235i·18-s − 0.458·19-s + 0.436·21-s + 1.25i·23-s − 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7972324763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7972324763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.230611402726732812170573677288, −8.657938088058476386904902920214, −7.84996991225705324931273227250, −7.22842422311618937493395305917, −6.32073832279531656660949266772, −5.68684127396678249394759724577, −4.94913320452898524656789408510, −3.75096483742186052697352024645, −2.68121323463964597993330153684, −1.45010747657485842520841235924,
0.28860504798558845416016847759, 1.79554827177164743504482470586, 2.94582154192333387566604590077, 3.91828874944976649504861745303, 4.49121222152920998992895410726, 5.40890418722005765692201476575, 6.46494201568305731324293918266, 7.32696566167836058548444849898, 8.346502302177859191431134507207, 8.951924251777135754726001946533
