L(s) = 1 | − 1.23i·2-s − 0.363i·3-s + 0.477·4-s + (1.29 − 1.82i)5-s − 0.449·6-s − 2.58i·7-s − 3.05i·8-s + 2.86·9-s + (−2.24 − 1.59i)10-s − 0.173i·12-s + 2.75i·13-s − 3.19·14-s + (−0.663 − 0.471i)15-s − 2.81·16-s + 3.85i·17-s − 3.53i·18-s + ⋯ |
L(s) = 1 | − 0.872i·2-s − 0.210i·3-s + 0.238·4-s + (0.579 − 0.815i)5-s − 0.183·6-s − 0.977i·7-s − 1.08i·8-s + 0.955·9-s + (−0.711 − 0.505i)10-s − 0.0501i·12-s + 0.765i·13-s − 0.852·14-s + (−0.171 − 0.121i)15-s − 0.704·16-s + 0.934i·17-s − 0.834i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.894709 - 1.73306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.894709 - 1.73306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.29 + 1.82i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.23iT - 2T^{2} \) |
| 3 | \( 1 + 0.363iT - 3T^{2} \) |
| 7 | \( 1 + 2.58iT - 7T^{2} \) |
| 13 | \( 1 - 2.75iT - 13T^{2} \) |
| 17 | \( 1 - 3.85iT - 17T^{2} \) |
| 19 | \( 1 - 0.277T + 19T^{2} \) |
| 23 | \( 1 - 8.40iT - 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 - 0.564T + 31T^{2} \) |
| 37 | \( 1 + 0.522iT - 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 + 2.54iT - 43T^{2} \) |
| 47 | \( 1 + 4.92iT - 47T^{2} \) |
| 53 | \( 1 - 8.72iT - 53T^{2} \) |
| 59 | \( 1 + 7.50T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 3.20iT - 67T^{2} \) |
| 71 | \( 1 + 8.40T + 71T^{2} \) |
| 73 | \( 1 - 13.0iT - 73T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 + 3.29iT - 83T^{2} \) |
| 89 | \( 1 + 2.48T + 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−10.20303708239544257011162051348, −9.864060598565904751559886639140, −8.856782796531237698143280228584, −7.53390828936231474825457478831, −6.89359321070953447080622269420, −5.77810554144044185715567980923, −4.37873508629654223912210624631, −3.67242698305647653730054350325, −1.91381353749165689576089090637, −1.22921144103181858156412248283,
2.11752355547260130492552762217, 3.05124605669666902601914186215, 4.81054446442392977909919129989, 5.68235575197049061236699311595, 6.51788121943812012223895944352, 7.20618651718318998507051364164, 8.155714797531000335615096882929, 9.162809901694595174913446861061, 10.09151023095482852601753571085, 10.81628581188652995255924211377