L(s) = 1 | + 2.09·5-s − i·7-s + 0.961i·11-s − 4.60i·13-s + 3.28i·17-s − 3.66·19-s − 2.40·23-s − 0.618·25-s + 9.31·29-s + 1.84i·31-s − 2.09i·35-s − 2.29i·37-s − 0.314i·41-s + 8.64·43-s − 2.34·47-s + ⋯ |
L(s) = 1 | + 0.936·5-s − 0.377i·7-s + 0.289i·11-s − 1.27i·13-s + 0.795i·17-s − 0.841·19-s − 0.501·23-s − 0.123·25-s + 1.73·29-s + 0.332i·31-s − 0.353i·35-s − 0.377i·37-s − 0.0491i·41-s + 1.31·43-s − 0.341·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.200799244\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.200799244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 2.09T + 5T^{2} \) |
| 11 | \( 1 - 0.961iT - 11T^{2} \) |
| 13 | \( 1 + 4.60iT - 13T^{2} \) |
| 17 | \( 1 - 3.28iT - 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 - 9.31T + 29T^{2} \) |
| 31 | \( 1 - 1.84iT - 31T^{2} \) |
| 37 | \( 1 + 2.29iT - 37T^{2} \) |
| 41 | \( 1 + 0.314iT - 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 + 2.34T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 + 7.99iT - 59T^{2} \) |
| 61 | \( 1 + 10.6iT - 61T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 1.26iT - 79T^{2} \) |
| 83 | \( 1 - 2.99iT - 83T^{2} \) |
| 89 | \( 1 + 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 4.98T + 97T^{2} \) |
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
−8.164932635071784669116394883971, −7.21468940240029787969288124455, −6.40614368750036677126924379559, −5.92204297257466301204403669582, −5.16521955066987588302681661868, −4.35114196366469723722584722260, −3.49766532271911128941500976366, −2.53113943323709258571392503020, −1.76823993177447260918362245472, −0.60825017278081078169402014696,
1.03530802913198646808129273590, 2.18833129725820534463237532869, 2.59684579988527375831143898320, 3.86746490521037229307382690061, 4.59432881404480745219022767366, 5.37300249865623945566725264149, 6.20606136766625835636699289604, 6.53371015910671190176901722000, 7.42105685860072651665249768156, 8.311164202483921687771936443211