L(s) = 1 | − 2-s + (−0.522 − 1.65i)3-s + 4-s + (−1.93 + 1.11i)5-s + (0.522 + 1.65i)6-s + 3.41i·7-s − 8-s + (−2.45 + 1.72i)9-s + (1.93 − 1.11i)10-s − 3.28·11-s + (−0.522 − 1.65i)12-s − 4.68·13-s − 3.41i·14-s + (2.85 + 2.61i)15-s + 16-s − 5.44i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.301 − 0.953i)3-s + 0.5·4-s + (−0.866 + 0.500i)5-s + (0.213 + 0.674i)6-s + 1.29i·7-s − 0.353·8-s + (−0.817 + 0.575i)9-s + (0.612 − 0.353i)10-s − 0.991·11-s + (−0.150 − 0.476i)12-s − 1.29·13-s − 0.913i·14-s + (0.738 + 0.674i)15-s + 0.250·16-s − 1.32i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.478582 - 0.301267i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.478582 - 0.301267i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.522 + 1.65i)T \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| 31 | \( 1 + (5.16 + 2.08i)T \) |
good | 7 | \( 1 - 3.41iT - 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 17 | \( 1 + 5.44iT - 17T^{2} \) |
| 19 | \( 1 - 6.86T + 19T^{2} \) |
| 23 | \( 1 + 3.10iT - 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 37 | \( 1 - 9.62T + 37T^{2} \) |
| 41 | \( 1 + 2.33iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 8.08T + 47T^{2} \) |
| 53 | \( 1 - 6.10iT - 53T^{2} \) |
| 59 | \( 1 + 10.9iT - 59T^{2} \) |
| 61 | \( 1 + 4.84iT - 61T^{2} \) |
| 67 | \( 1 + 1.08iT - 67T^{2} \) |
| 71 | \( 1 - 1.09iT - 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 1.97iT - 79T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
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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
−9.821353387999068462018372298196, −9.012972106804915865466050225871, −7.945756579811708911159788176250, −7.56061750358220820785205821180, −6.82063735646636894810326971399, −5.67117667345400127927550108882, −4.92853313803562638157617710435, −2.72980481625479875017409067249, −2.58340040488761921082457338254, −0.48771478188260169094709592799,
0.866427365263877521031651010731, 2.97258075781809445009684628907, 3.99732786014289173786873405009, 4.82291474291368708575574752754, 5.76193764632302721572220945423, 7.27300730634028661256348961483, 7.64573498775830073086671239066, 8.577549311432656747252792278503, 9.582268832963226199017122389462, 10.20340093508750279481680738445