| L(s) = 1 | + i·5-s − 0.585i·7-s + 1.41·11-s + 2.24·13-s + 1.17i·17-s − 5.65i·19-s + 3.17·23-s − 25-s + 2i·29-s + 3.17i·31-s + 0.585·35-s + 4.58·37-s + 8.24i·41-s − 6.82i·43-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.221i·7-s + 0.426·11-s + 0.621·13-s + 0.284i·17-s − 1.29i·19-s + 0.661·23-s − 0.200·25-s + 0.371i·29-s + 0.569i·31-s + 0.0990·35-s + 0.753·37-s + 1.28i·41-s − 1.04i·43-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784918634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784918634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 + 0.585iT - 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 19 | \( 1 + 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 3.17iT - 31T^{2} \) |
| 37 | \( 1 - 4.58T + 37T^{2} \) |
| 41 | \( 1 - 8.24iT - 41T^{2} \) |
| 43 | \( 1 + 6.82iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6.82iT - 53T^{2} \) |
| 59 | \( 1 - 5.41T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 0.828T + 73T^{2} \) |
| 79 | \( 1 - 6.48iT - 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 - 0.928iT - 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.409629576727374861772751571554, −8.878561476747410254777922117202, −7.915386712093508364303447892688, −7.00046140158803314547142667568, −6.45313737926219890746712916614, −5.42345447247360433116709228138, −4.41483979794490151104852522025, −3.48337194332853357421615833637, −2.47852701520082695114679370995, −1.02624284761242937505374598830,
0.989747036015787197662246784592, 2.25167325524356574603737778445, 3.57074414181860028615381546493, 4.34466011803494017218922840170, 5.48177391942134088232947014721, 6.09090468854265338273754895294, 7.11505137958630857719277112864, 8.006519077197250871754554743353, 8.726804316704967158692421844840, 9.435819829709387092592822241179
