| L(s) = 1 | + (−0.408 − 0.706i)2-s + (34.4 + 59.7i)3-s + (63.6 − 110. i)4-s − 447.·5-s + (28.1 − 48.7i)6-s + (171.5 − 297. i)7-s − 208.·8-s + (−1.28e3 + 2.22e3i)9-s + (182. + 315. i)10-s + (2.65e3 + 4.59e3i)11-s + 8.77e3·12-s + (−2.74e3 − 7.43e3i)13-s − 279.·14-s + (−1.54e4 − 2.66e4i)15-s + (−8.06e3 − 1.39e4i)16-s + (2.52e3 − 4.38e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0360 − 0.0624i)2-s + (0.737 + 1.27i)3-s + (0.497 − 0.861i)4-s − 1.59·5-s + (0.0531 − 0.0920i)6-s + (0.188 − 0.327i)7-s − 0.143·8-s + (−0.586 + 1.01i)9-s + (0.0576 + 0.0999i)10-s + (0.600 + 1.04i)11-s + 1.46·12-s + (−0.346 − 0.938i)13-s − 0.0272·14-s + (−1.17 − 2.04i)15-s + (−0.492 − 0.852i)16-s + (0.124 − 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.53946 - 0.790978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53946 - 0.790978i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (2.74e3 + 7.43e3i)T \) |
| good | 2 | \( 1 + (0.408 + 0.706i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-34.4 - 59.7i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 447.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (-2.65e3 - 4.59e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-2.52e3 + 4.38e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.00e4 + 3.47e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.27e4 + 3.94e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (7.56e4 + 1.30e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-9.92e4 - 1.71e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (1.43e5 + 2.48e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.44e5 + 5.97e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 3.06e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.22e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-5.82e5 + 1.00e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-3.64e5 + 6.30e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.13e6 + 1.96e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.60e6 - 4.50e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 6.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (5.57e5 + 9.65e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.75e5 - 3.04e5i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−12.17648086549840836248177510718, −11.25693536003097648643013958523, −10.26073269431082379389456402053, −9.438431653183138246773750013771, −8.086211221637153621382486930910, −7.01195616357898855117500528771, −4.92803652019577861979235615495, −4.09052228222735155175145809011, −2.75469355460911955055590412330, −0.54053323714876447399368934080,
1.36934223147349941258279553293, 2.96360997052246649006962257857, 3.92884659704079237774887413297, 6.41978414694987727146710406448, 7.54217355596967768981632335464, 8.011807534674314024127766411970, 8.910115776053831914290107602350, 11.33236942476014979585856850242, 11.91317359315963517443356901530, 12.56333896332432186361009826155
