L(s) = 1 | + (−1.73 + i)2-s + (5.10 + 0.957i)3-s + (1.99 − 3.46i)4-s + (−0.0529 + 0.0917i)5-s + (−9.80 + 3.44i)6-s + (−7.55 + 16.9i)7-s + 7.99i·8-s + (25.1 + 9.77i)9-s − 0.211i·10-s + (−23.9 + 13.8i)11-s + (13.5 − 15.7i)12-s + (54.2 + 31.3i)13-s + (−3.82 − 36.8i)14-s + (−0.358 + 0.418i)15-s + (−8 − 13.8i)16-s + 57.7·17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.982 + 0.184i)3-s + (0.249 − 0.433i)4-s + (−0.00473 + 0.00820i)5-s + (−0.667 + 0.234i)6-s + (−0.407 + 0.913i)7-s + 0.353i·8-s + (0.932 + 0.362i)9-s − 0.00670i·10-s + (−0.655 + 0.378i)11-s + (0.325 − 0.379i)12-s + (1.15 + 0.668i)13-s + (−0.0730 − 0.703i)14-s + (−0.00617 + 0.00719i)15-s + (−0.125 − 0.216i)16-s + 0.824·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0923 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0923 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.16710 + 1.06381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16710 + 1.06381i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 - i)T \) |
| 3 | \( 1 + (-5.10 - 0.957i)T \) |
| 7 | \( 1 + (7.55 - 16.9i)T \) |
good | 5 | \( 1 + (0.0529 - 0.0917i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (23.9 - 13.8i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-54.2 - 31.3i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 57.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (82.4 + 47.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (242. - 139. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-196. - 113. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 104.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-130. + 225. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (220. + 381. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (134. + 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 26.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-97.1 + 168. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-418. + 241. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-225. + 391. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 979. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 30.5iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-227. - 394. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-345. - 598. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 145.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-909. + 525. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.26904675944707793294648210763, −12.20230655858568994063513534559, −10.74839891174102790826502480546, −9.704584680774085180844186271513, −8.862247127679713844208140418556, −8.032345662702439227167451155659, −6.78373245552992972148260302410, −5.36625077481720437743038726222, −3.49041554862782951192623866229, −1.91842190562589931606823946975,
0.961584005583489253579337799376, 2.86206610213503731818999740025, 3.98416068428924686184838642599, 6.24235700185785435152245578735, 7.71272185888066805233509932013, 8.215977452328091628763206778332, 9.615529613862668562175431807831, 10.30389393678794461049402892493, 11.45540091406782779342758196499, 13.03483901661596958535824154708