| L(s) = 1 | + (−5.90 − 10.2i)2-s + (36.7 + 63.7i)3-s + (−5.68 + 9.84i)4-s − 76.4·5-s + (434. − 752. i)6-s + (171.5 − 297. i)7-s − 1.37e3·8-s + (−1.61e3 + 2.79e3i)9-s + (451. + 781. i)10-s + (−1.87e3 − 3.25e3i)11-s − 836.·12-s + (5.87e3 + 5.31e3i)13-s − 4.04e3·14-s + (−2.81e3 − 4.87e3i)15-s + (8.85e3 + 1.53e4i)16-s + (−1.59e4 + 2.75e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.521 − 0.903i)2-s + (0.786 + 1.36i)3-s + (−0.0443 + 0.0769i)4-s − 0.273·5-s + (0.820 − 1.42i)6-s + (0.188 − 0.327i)7-s − 0.950·8-s + (−0.737 + 1.27i)9-s + (0.142 + 0.247i)10-s + (−0.425 − 0.737i)11-s − 0.139·12-s + (0.741 + 0.671i)13-s − 0.394·14-s + (−0.215 − 0.372i)15-s + (0.540 + 0.936i)16-s + (−0.785 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0468686 + 0.196648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0468686 + 0.196648i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 + 297. i)T \) |
| 13 | \( 1 + (-5.87e3 - 5.31e3i)T \) |
| good | 2 | \( 1 + (5.90 + 10.2i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-36.7 - 63.7i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 76.4T + 7.81e4T^{2} \) |
| 11 | \( 1 + (1.87e3 + 3.25e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.59e4 - 2.75e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-5.67e3 + 9.82e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.17e4 + 3.77e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (8.80e4 + 1.52e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.52e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.44e5 - 2.50e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (2.85e5 + 4.93e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.30e5 - 7.44e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 3.21e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.72e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (9.03e5 - 1.56e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-9.94e5 + 1.72e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.79e5 - 4.83e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-7.65e5 + 1.32e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 2.55e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.67e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.80e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (1.69e6 + 2.92e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-4.69e6 + 8.13e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.14790491704491914968756712263, −11.43068230951169908198269171202, −10.85221169615168352968351471089, −9.961264584747496308977984159838, −8.975065664040569008934648002493, −8.205519091463777251062636336093, −6.04373393870045605089141255230, −4.24689207932188783764980675208, −3.34924508136007415265841524470, −1.89740931796092880185799739401,
0.06335487153557608585256552288, 1.87088201689817856282961236577, 3.21647915548635750003867742792, 5.62126187462864016895820080237, 7.01986725095642036667422399296, 7.57820988745645199518537750577, 8.448822184241346193238251021869, 9.410100947419966773314407328152, 11.40796504791567651653018506618, 12.46029296838071274182812787868
