| L(s) = 1 | + (−11.1 − 1.87i)2-s + (26.2 − 63.3i)3-s + (120. + 41.9i)4-s + (−452. + 187. i)5-s + (−411. + 657. i)6-s + (210. − 210. i)7-s + (−1.27e3 − 694. i)8-s + (−1.78e3 − 1.78e3i)9-s + (5.40e3 − 1.24e3i)10-s + (1.85e3 + 4.47e3i)11-s + (5.83e3 − 6.56e3i)12-s + (−6.54e3 − 2.71e3i)13-s + (−2.74e3 + 1.95e3i)14-s + 3.36e4i·15-s + (1.28e4 + 1.01e4i)16-s + 2.58e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.986 − 0.165i)2-s + (0.561 − 1.35i)3-s + (0.944 + 0.327i)4-s + (−1.61 + 0.670i)5-s + (−0.778 + 1.24i)6-s + (0.232 − 0.232i)7-s + (−0.877 − 0.479i)8-s + (−0.814 − 0.814i)9-s + (1.70 − 0.392i)10-s + (0.420 + 1.01i)11-s + (0.974 − 1.09i)12-s + (−0.826 − 0.342i)13-s + (−0.267 + 0.190i)14-s + 2.57i·15-s + (0.785 + 0.618i)16-s + 1.27i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.344940 + 0.269383i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.344940 + 0.269383i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.1 + 1.87i)T \) |
| good | 3 | \( 1 + (-26.2 + 63.3i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (452. - 187. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-210. + 210. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-1.85e3 - 4.47e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (6.54e3 + 2.71e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 2.58e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-2.06e4 - 8.54e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-2.44e4 - 2.44e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (8.01e4 - 1.93e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.39e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (4.89e5 - 2.02e5i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (1.76e5 + 1.76e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (1.27e5 + 3.08e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 8.19e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-6.85e4 - 1.65e5i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (2.11e6 - 8.77e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (2.62e5 - 6.34e5i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (9.76e5 - 2.35e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-2.66e5 + 2.66e5i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (1.90e6 + 1.90e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 5.03e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (3.53e6 + 1.46e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (2.09e6 - 2.09e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 1.58e7T + 8.07e13T^{2} \) |
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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
−15.37537085129755052777016794491, −14.59979240095253394056038601422, −12.55771064834442438947530398981, −11.92582201300569596537060250913, −10.53513944170714943338257283215, −8.584941951326571155204145646718, −7.38681391681071225504827968539, −7.14060152999506846877062761145, −3.37358799263302936810697569328, −1.60376873096526308669110366128,
0.26743088409422034224175968619, 3.29429058390157986485809023692, 4.88419189314110555620326324587, 7.51779577307247033624652019147, 8.706995212276799639157074597321, 9.461315534117014620276307842049, 11.12770600810371419696881033655, 11.89411879519973135662469662334, 14.43085413944768692710021978150, 15.48302250341543200970380001759
