L(s) = 1 | − 1.41i·5-s + 6·13-s − 4.24i·17-s + 2.99·25-s + 9.89i·29-s − 12·37-s − 12.7i·41-s + 7·49-s − 7.07i·53-s + 12·61-s − 8.48i·65-s + 16·73-s − 6·85-s − 4.24i·89-s + 8·97-s + ⋯ |
L(s) = 1 | − 0.632i·5-s + 1.66·13-s − 1.02i·17-s + 0.599·25-s + 1.83i·29-s − 1.97·37-s − 1.98i·41-s + 49-s − 0.971i·53-s + 1.53·61-s − 1.05i·65-s + 1.87·73-s − 0.650·85-s − 0.449i·89-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.866192474\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.866192474\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 12T + 37T^{2} \) |
| 41 | \( 1 + 12.7iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.665448327469721039064885177842, −8.520252027462985877460482464529, −7.18987703843219766734943236132, −6.71844592878880321149052682834, −5.50993757594403209479604473140, −5.10944623049805629349569752524, −3.92845347855090775718477750503, −3.22571441015524043697539923249, −1.83646560878485289233767997593, −0.75773277705540921345974271839,
1.16226903652818872137226635763, 2.36249069848300583001755989159, 3.48144293144950964823622845660, 4.06305588673099705790410299508, 5.25931563034502464343249746713, 6.26246697919264591378252245852, 6.53037306920614621635227014891, 7.68150653352053951624202895369, 8.358302792435375332841741510610, 8.991704092772574245798637408766