| L(s) = 1 | + (13.7 − 5.01i)3-s + (38.0 − 216. i)5-s + (−872. − 1.51e3i)7-s + (−1.51e3 + 1.26e3i)9-s + (−3.05e3 + 5.28e3i)11-s + (1.08e4 + 3.96e3i)13-s + (−558. − 3.16e3i)15-s + (−2.77e3 − 2.32e3i)17-s + (−2.90e4 + 7.13e3i)19-s + (−1.95e4 − 1.64e4i)21-s + (4.93e3 + 2.80e4i)23-s + (2.82e4 + 1.02e4i)25-s + (−3.04e4 + 5.27e4i)27-s + (−9.45e4 + 7.93e4i)29-s + (1.17e4 + 2.03e4i)31-s + ⋯ |
| L(s) = 1 | + (0.294 − 0.107i)3-s + (0.136 − 0.772i)5-s + (−0.961 − 1.66i)7-s + (−0.690 + 0.579i)9-s + (−0.691 + 1.19i)11-s + (1.37 + 0.500i)13-s + (−0.0427 − 0.242i)15-s + (−0.137 − 0.115i)17-s + (−0.971 + 0.238i)19-s + (−0.461 − 0.387i)21-s + (0.0846 + 0.480i)23-s + (0.361 + 0.131i)25-s + (−0.298 + 0.516i)27-s + (−0.719 + 0.604i)29-s + (0.0707 + 0.122i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.711i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0590951 + 0.141515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0590951 + 0.141515i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.90e4 - 7.13e3i)T \) |
| good | 3 | \( 1 + (-13.7 + 5.01i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-38.0 + 216. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (872. + 1.51e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.05e3 - 5.28e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.08e4 - 3.96e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (2.77e3 + 2.32e3i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-4.93e3 - 2.80e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (9.45e4 - 7.93e4i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-1.17e4 - 2.03e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.69e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (5.25e5 - 1.91e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-4.09e4 + 2.32e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (2.92e5 - 2.45e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (2.08e5 + 1.18e6i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (1.88e6 + 1.58e6i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (3.55e5 + 2.01e6i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.92e6 - 1.61e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (5.92e4 - 3.35e5i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-1.68e6 + 6.12e5i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (2.19e6 - 7.98e5i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (1.47e6 + 2.54e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.40e6 + 1.23e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-4.94e6 - 4.14e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−13.25250176543790147478697589802, −12.89063033209811975460336530474, −11.08406435648739572267561645443, −10.15140359315419603200770001378, −8.938748506495735799902901556803, −7.71358847364895957455511446729, −6.53972962070782731699462100129, −4.82758038649881927336623835764, −3.53478774359705918094892311615, −1.59968534039279294580758463584,
0.04770025542776194999089517946, 2.63173710851040766339134591179, 3.32195638264842025942563193280, 5.85504399640440681872354671746, 6.26925948206509169384282569866, 8.439075511076806290403097418848, 8.979472401285375929720669815157, 10.49154237059424078543989691065, 11.48435713949065794167705976579, 12.75058365676811865586518039688
