| L(s) = 1 | − 1.77i·2-s − 1.13·4-s + (−1.51 + 1.64i)5-s − 0.437i·7-s − 1.52i·8-s + (2.92 + 2.67i)10-s − 5.73·11-s − i·13-s − 0.775·14-s − 4.98·16-s − 3.98i·17-s − 4.77·19-s + (1.71 − 1.87i)20-s + 10.1i·22-s + 0.337i·23-s + ⋯ |
| L(s) = 1 | − 1.25i·2-s − 0.569·4-s + (−0.675 + 0.737i)5-s − 0.165i·7-s − 0.539i·8-s + (0.923 + 0.846i)10-s − 1.72·11-s − 0.277i·13-s − 0.207·14-s − 1.24·16-s − 0.965i·17-s − 1.09·19-s + (0.384 − 0.419i)20-s + 2.16i·22-s + 0.0704i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.157528 + 0.405195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157528 + 0.405195i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.51 - 1.64i)T \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 1.77iT - 2T^{2} \) |
| 7 | \( 1 + 0.437iT - 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 17 | \( 1 + 3.98iT - 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 - 0.337iT - 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 - 1.66iT - 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 + 2.27iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 + 14.2iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 - 2.17iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 2.55iT - 73T^{2} \) |
| 79 | \( 1 + 2.55T + 79T^{2} \) |
| 83 | \( 1 + 5.41iT - 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 + 14.7iT - 97T^{2} \) |
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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
−10.47597055341175659779051671383, −9.725742005701114425839986470878, −8.443075226026666176738667773195, −7.51736319018883812211113796353, −6.72217139185915522241961681703, −5.26353351293546592869986382209, −4.08626943590281925009779830638, −3.04513961789868960090596076485, −2.29131115901805935767143992781, −0.22327580855592026506589675468,
2.26164185650448820336211566273, 3.99428329176277437070848944440, 5.08470651490846297376220529511, 5.71422492005432514700772915929, 6.87329878849980966558754179962, 7.76822978997190159748696675003, 8.339639955696894232828632978670, 9.007578250105764165924242602732, 10.43750332223598692801392553913, 11.11956406907534140906298587973
