| L(s) = 1 | + (1.23 − 3.80i)2-s + (−16.6 − 12.0i)3-s + (−12.9 − 9.40i)4-s + (2.13 − 55.8i)5-s + (−66.4 + 48.2i)6-s + 12.6·7-s + (−51.7 + 37.6i)8-s + (55.1 + 169. i)9-s + (−209. − 77.1i)10-s + (−169. + 521. i)11-s + (101. + 312. i)12-s + (−13.3 − 41.0i)13-s + (15.6 − 48.1i)14-s + (−709. + 901. i)15-s + (79.1 + 243. i)16-s + (−175. + 127. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−1.06 − 0.774i)3-s + (−0.404 − 0.293i)4-s + (0.0382 − 0.999i)5-s + (−0.753 + 0.547i)6-s + 0.0975·7-s + (−0.286 + 0.207i)8-s + (0.226 + 0.698i)9-s + (−0.663 − 0.244i)10-s + (−0.422 + 1.30i)11-s + (0.203 + 0.626i)12-s + (−0.0219 − 0.0674i)13-s + (0.0213 − 0.0655i)14-s + (−0.814 + 1.03i)15-s + (0.0772 + 0.237i)16-s + (−0.146 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.247777 + 0.412531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247777 + 0.412531i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 5 | \( 1 + (-2.13 + 55.8i)T \) |
| good | 3 | \( 1 + (16.6 + 12.0i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 12.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + (169. - 521. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (13.3 + 41.0i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (175. - 127. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-135. + 98.5i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-1.12e3 + 3.47e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (5.13e3 + 3.72e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (5.35e3 - 3.88e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (1.48e3 + 4.55e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.67e3 + 8.22e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.79e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.21e3 + 4.51e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.11e4 + 8.11e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.49e4 - 4.58e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (639. - 1.96e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (4.28e4 - 3.11e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (2.72e4 + 1.98e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.20e3 - 3.72e3i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (5.87e4 + 4.26e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-7.38e4 + 5.36e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.40e4 + 1.04e5i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-8.27e4 - 6.01e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.08864232005122026087072025905, −12.60873278498889803987383210804, −11.73959358493530312514062126814, −10.48547754658178847715393412660, −9.051150121309469666906925933007, −7.34037593128005205316817036269, −5.70669476596881471582186852404, −4.55568700703941561014982510475, −1.80659277525083112874668055002, −0.25780314626981872101926054344,
3.50004779266540663843993086750, 5.26755295615576981344492388606, 6.19440307860374709493264533930, 7.67762370809436490187771211985, 9.477192486402831198269935786927, 10.88218030012089106192563694789, 11.42077899934889126237400227219, 13.24169296037178332612387669730, 14.39756407010621426642996923274, 15.50148413966640980175016195949
