| L(s) = 1 | + (116. − 202. i)5-s + (−84.3 − 903. i)7-s + (−778. + 449. i)11-s + 6.65e3i·13-s + (1.20e4 + 2.09e4i)17-s + (1.31e4 + 7.58e3i)19-s + (4.56e4 + 2.63e4i)23-s + (1.18e4 + 2.05e4i)25-s + 3.62e4i·29-s + (1.31e5 − 7.56e4i)31-s + (−1.92e5 − 8.83e4i)35-s + (3.06e4 − 5.30e4i)37-s − 2.66e5·41-s + 3.23e5·43-s + (3.52e5 − 6.09e5i)47-s + ⋯ |
| L(s) = 1 | + (0.417 − 0.722i)5-s + (−0.0929 − 0.995i)7-s + (−0.176 + 0.101i)11-s + 0.840i·13-s + (0.596 + 1.03i)17-s + (0.439 + 0.253i)19-s + (0.782 + 0.451i)23-s + (0.151 + 0.262i)25-s + 0.275i·29-s + (0.790 − 0.456i)31-s + (−0.758 − 0.348i)35-s + (0.0994 − 0.172i)37-s − 0.602·41-s + 0.619·43-s + (0.494 − 0.856i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.421707736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.421707736\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (84.3 + 903. i)T \) |
| good | 5 | \( 1 + (-116. + 202. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (778. - 449. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 6.65e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.20e4 - 2.09e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.31e4 - 7.58e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.56e4 - 2.63e4i)T + (1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 3.62e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.31e5 + 7.56e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-3.06e4 + 5.30e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.66e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.52e5 + 6.09e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-6.24e5 + 3.60e5i)T + (5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (8.13e4 + 1.40e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.14e6 + 6.59e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.74e6 - 3.01e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 3.93e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.29e6 + 7.45e5i)T + (5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (5.09e5 - 8.83e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 1.08e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.21e6 + 7.30e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 3.46e6iT - 8.07e13T^{2} \) |
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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
−10.61060088055948102611799106291, −9.780829879875516391662480342249, −8.872158057187969162535914509445, −7.77380229757933012894364318187, −6.79034836218760961934260435843, −5.59341809759198064404040237858, −4.52531887230968603217056251814, −3.45940327895202494053923148318, −1.74764362884211710937530660187, −0.815627264133188656061681878104,
0.813885557592802965463263884743, 2.52355683956879776850884567225, 3.08579742006466132916765471644, 4.92565554462580401884882228345, 5.81769380681922821888498914134, 6.82304717772476062690606048032, 7.936002453567447534300755259444, 9.023412349482066246891972795643, 9.935902218435212212272371100157, 10.80343949842778124034256872832
