| L(s) = 1 | + (9.76 − 16.9i)2-s + (43.0 − 74.4i)3-s + (−126. − 219. i)4-s + 20.2·5-s + (−839. − 1.45e3i)6-s + (171.5 + 297. i)7-s − 2.44e3·8-s + (−2.60e3 − 4.51e3i)9-s + (197. − 341. i)10-s + (−1.46e3 + 2.53e3i)11-s − 2.17e4·12-s + (5.13e3 + 6.02e3i)13-s + 6.69e3·14-s + (869. − 1.50e3i)15-s + (−7.68e3 + 1.33e4i)16-s + (−1.27e4 − 2.20e4i)17-s + ⋯ |
| L(s) = 1 | + (0.863 − 1.49i)2-s + (0.919 − 1.59i)3-s + (−0.989 − 1.71i)4-s + 0.0723·5-s + (−1.58 − 2.74i)6-s + (0.188 + 0.327i)7-s − 1.68·8-s + (−1.19 − 2.06i)9-s + (0.0623 − 0.108i)10-s + (−0.331 + 0.574i)11-s − 3.64·12-s + (0.648 + 0.761i)13-s + 0.652·14-s + (0.0664 − 0.115i)15-s + (−0.468 + 0.812i)16-s + (−0.628 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.01492 + 3.16685i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01492 + 3.16685i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-5.13e3 - 6.02e3i)T \) |
| good | 2 | \( 1 + (-9.76 + 16.9i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (-43.0 + 74.4i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 - 20.2T + 7.81e4T^{2} \) |
| 11 | \( 1 + (1.46e3 - 2.53e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.27e4 + 2.20e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.71e3 - 6.44e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (9.24e3 - 1.60e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-8.58e4 + 1.48e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.62e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.76e5 + 3.04e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-2.56e5 + 4.44e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.36e4 + 4.09e4i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + 2.61e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.40e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.37e6 - 2.37e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.01e6 - 1.75e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.48e5 - 1.29e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (7.94e5 + 1.37e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 - 2.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.11e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.41e6 + 4.18e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (8.05e6 + 1.39e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−11.97453580146113561254685419205, −11.57338121591713222974771373529, −9.837695047437432580827809994413, −8.733352124267141214704439227431, −7.37403959122789505821116350209, −5.96051124766446672392106978502, −4.18325380407744255735003096863, −2.65797505536526139097397288872, −2.02936835647902126796225440416, −0.807979279925463939966896063812,
3.11725679624680186305275583019, 4.12049431984018316375411091931, 5.05156031112952248666745002957, 6.25948652168532939395122266840, 8.146301328678988540334359671260, 8.420209896376957668579770808555, 9.964131114100035124139614461713, 11.01810778807059629208145408177, 13.08764268620638192747328292716, 13.82460351246867467635218932327
