| L(s) = 1 | + 14.0·2-s + (−5.97 + 10.3i)3-s + 68.7·4-s + (−104. + 181. i)5-s + (−83.7 + 145. i)6-s + (163. + 282. i)7-s − 831.·8-s + (1.02e3 + 1.77e3i)9-s + (−1.47e3 + 2.54e3i)10-s − 5.35e3·11-s + (−410. + 710. i)12-s + (4.67e3 + 8.09e3i)13-s + (2.28e3 + 3.96e3i)14-s + (−1.25e3 − 2.17e3i)15-s − 2.04e4·16-s + (3.18e3 + 5.51e3i)17-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + (−0.127 + 0.221i)3-s + 0.536·4-s + (−0.375 + 0.650i)5-s + (−0.158 + 0.274i)6-s + (0.179 + 0.311i)7-s − 0.574·8-s + (0.467 + 0.809i)9-s + (−0.465 + 0.806i)10-s − 1.21·11-s + (−0.0685 + 0.118i)12-s + (0.589 + 1.02i)13-s + (0.222 + 0.386i)14-s + (−0.0958 − 0.166i)15-s − 1.24·16-s + (0.157 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.23133 + 1.90610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23133 + 1.90610i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.79e5 + 2.05e5i)T \) |
| good | 2 | \( 1 - 14.0T + 128T^{2} \) |
| 3 | \( 1 + (5.97 - 10.3i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (104. - 181. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-163. - 282. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + 5.35e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.67e3 - 8.09e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-3.18e3 - 5.51e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.05e4 - 1.82e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-5.42e4 + 9.39e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.01e4 - 3.48e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-8.49e4 + 1.47e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.41e5 - 4.17e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 5.82e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 7.86e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-8.73e5 + 1.51e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + 1.21e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-2.37e5 - 4.11e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.57e6 - 2.72e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.32e6 - 4.01e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-7.72e5 - 1.33e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.28e5 - 3.95e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.78e6 - 6.55e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.08e6 - 5.34e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.50e6T + 8.07e13T^{2} \) |
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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
−14.76084124973445982755729706566, −13.63532283130063769766727739515, −12.70655535319326264201552255849, −11.40081512761735277723229425797, −10.37963551534940037385122504121, −8.450977752852961210992339235841, −6.79851860966498737150323593402, −5.30155690303465642662932077113, −4.13003592457881034264054491713, −2.53665841424292517499303463158,
0.65125364470103533425985540727, 3.20251500124173609732320617486, 4.62610750895252822880905865189, 5.77398298067106150437989232910, 7.46277233925288665493055477248, 9.017756785880466779693805080979, 10.79289733472701283793669816662, 12.22364963675980059710065101478, 12.92068277850169541530261846772, 13.76816786327809635342956361513
