L(s) = 1 | − 4.89·5-s + 1.73·7-s − 8.48·11-s − 5.19i·13-s + 25.4i·17-s − 5i·19-s + 24.4i·23-s − 1.00·25-s + 39.1·29-s + 13.8·31-s − 8.48·35-s − 32.9i·37-s − 16.9i·41-s − 34i·43-s − 4.89i·47-s + ⋯ |
L(s) = 1 | − 0.979·5-s + 0.247·7-s − 0.771·11-s − 0.399i·13-s + 1.49i·17-s − 0.263i·19-s + 1.06i·23-s − 0.0400·25-s + 1.35·29-s + 0.446·31-s − 0.242·35-s − 0.889i·37-s − 0.413i·41-s − 0.790i·43-s − 0.104i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9757517295\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9757517295\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4.89T + 25T^{2} \) |
| 7 | \( 1 - 1.73T + 49T^{2} \) |
| 11 | \( 1 + 8.48T + 121T^{2} \) |
| 13 | \( 1 + 5.19iT - 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 + 5iT - 361T^{2} \) |
| 23 | \( 1 - 24.4iT - 529T^{2} \) |
| 29 | \( 1 - 39.1T + 841T^{2} \) |
| 31 | \( 1 - 13.8T + 961T^{2} \) |
| 37 | \( 1 + 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 4.89iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.79T + 2.80e3T^{2} \) |
| 59 | \( 1 + 76.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 22.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 19iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 68.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 25T + 5.32e3T^{2} \) |
| 79 | \( 1 - 140.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 101.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 8.48iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 71T + 9.40e3T^{2} \) |
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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
−8.786597881873760699631771147035, −7.991477550959438653068790712741, −7.70553871359926325891019307004, −6.60869506195646179659090560679, −5.68717776437949248213721052017, −4.79350675207790913451984841868, −3.89872806327782644084531621279, −3.08552694168623057054353920629, −1.79026886765337345102507486722, −0.33531751018756129309494605214,
0.878000873276693675359906249040, 2.47025765612019520472340109294, 3.29046534992190201925510319570, 4.56680389321421659165323121385, 4.86904145657965830773149063034, 6.20698161499011140794945530866, 6.99249097913292150461961532941, 7.890976559328300568810461722021, 8.262800129742192151822247821295, 9.276717076147600964164376255056