| L(s) = 1 | + (−2.47 + 7.60i)2-s + (−20.1 − 4.28i)3-s + (−51.7 − 37.6i)4-s + (104. + 181. i)5-s + (82.4 − 142. i)6-s + (−243. + 108. i)7-s + (414. − 300. i)8-s + (−1.60e3 − 716. i)9-s + (−1.64e3 + 348. i)10-s + (−321. + 3.05e3i)11-s + (883. + 980. i)12-s + (−1.39e3 + 1.54e3i)13-s + (−222. − 2.11e3i)14-s + (−1.33e3 − 4.11e3i)15-s + (1.26e3 + 3.89e3i)16-s + (−1.26e3 − 1.20e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.431 − 0.0916i)3-s + (−0.404 − 0.293i)4-s + (0.375 + 0.649i)5-s + (0.155 − 0.270i)6-s + (−0.267 + 0.119i)7-s + (0.286 − 0.207i)8-s + (−0.735 − 0.327i)9-s + (−0.518 + 0.110i)10-s + (−0.0727 + 0.692i)11-s + (0.147 + 0.163i)12-s + (−0.175 + 0.195i)13-s + (−0.0216 − 0.206i)14-s + (−0.102 − 0.314i)15-s + (0.0772 + 0.237i)16-s + (−0.0622 − 0.592i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.610874 - 0.313158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.610874 - 0.313158i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.47 - 7.60i)T \) |
| 31 | \( 1 + (-1.57e5 + 5.08e4i)T \) |
| good | 3 | \( 1 + (20.1 + 4.28i)T + (1.99e3 + 889. i)T^{2} \) |
| 5 | \( 1 + (-104. - 181. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (243. - 108. i)T + (5.51e5 - 6.12e5i)T^{2} \) |
| 11 | \( 1 + (321. - 3.05e3i)T + (-1.90e7 - 4.05e6i)T^{2} \) |
| 13 | \( 1 + (1.39e3 - 1.54e3i)T + (-6.55e6 - 6.24e7i)T^{2} \) |
| 17 | \( 1 + (1.26e3 + 1.20e4i)T + (-4.01e8 + 8.53e7i)T^{2} \) |
| 19 | \( 1 + (2.11e4 + 2.35e4i)T + (-9.34e7 + 8.88e8i)T^{2} \) |
| 23 | \( 1 + (-5.46e4 + 3.97e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-3.25e4 + 1.00e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 37 | \( 1 + (2.24e4 - 3.89e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-4.63e5 + 9.84e4i)T + (1.77e11 - 7.92e10i)T^{2} \) |
| 43 | \( 1 + (4.82e5 + 5.35e5i)T + (-2.84e10 + 2.70e11i)T^{2} \) |
| 47 | \( 1 + (3.51e5 + 1.08e6i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-3.53e4 - 1.57e4i)T + (7.86e11 + 8.72e11i)T^{2} \) |
| 59 | \( 1 + (1.65e6 + 3.51e5i)T + (2.27e12 + 1.01e12i)T^{2} \) |
| 61 | \( 1 + 3.62e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + (5.76e5 + 9.97e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.58e6 + 7.04e5i)T + (6.08e12 + 6.75e12i)T^{2} \) |
| 73 | \( 1 + (2.65e5 - 2.52e6i)T + (-1.08e13 - 2.29e12i)T^{2} \) |
| 79 | \( 1 + (-2.19e5 - 2.08e6i)T + (-1.87e13 + 3.99e12i)T^{2} \) |
| 83 | \( 1 + (-1.14e6 + 2.42e5i)T + (2.47e13 - 1.10e13i)T^{2} \) |
| 89 | \( 1 + (-1.45e5 - 1.05e5i)T + (1.36e13 + 4.20e13i)T^{2} \) |
| 97 | \( 1 + (2.90e6 + 2.11e6i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−13.57391201222832289434250746940, −12.24950033382721191085454803477, −10.97770854326624533305310452102, −9.808227044966632166232079198547, −8.633257809336958213742599575491, −6.99556124100096053169126674053, −6.24047121049096026465492483404, −4.79106223576834117589088942668, −2.63201899873343449807742756269, −0.31045140925999378560155293689,
1.25666514747916094556335240707, 3.09011642573216267464956817865, 4.87308787280451489899240482513, 6.09961204748198728863902942214, 8.115223725521711690784686442384, 9.130617808199882732059116750001, 10.43467550197699003110969943042, 11.31964268515449433129793530079, 12.55387903292353974240998003385, 13.38568535569611327413361964739
