| L(s) = 1 | + (2.53 − 3.18i)2-s + (2.05 − 0.309i)3-s + (3.43 + 15.0i)4-s + (−53.1 + 36.2i)5-s + (4.22 − 7.31i)6-s + (95.1 + 164. i)7-s + (173. + 83.7i)8-s + (−228. + 70.3i)9-s + (−19.5 + 261. i)10-s + (96.9 − 424. i)11-s + (11.6 + 29.8i)12-s + (13.6 + 182. i)13-s + (765. + 115. i)14-s + (−97.8 + 90.8i)15-s + (262. − 126. i)16-s + (1.59e3 + 1.08e3i)17-s + ⋯ |
| L(s) = 1 | + (0.448 − 0.562i)2-s + (0.131 − 0.0198i)3-s + (0.107 + 0.470i)4-s + (−0.951 + 0.648i)5-s + (0.0478 − 0.0829i)6-s + (0.733 + 1.27i)7-s + (0.960 + 0.462i)8-s + (−0.938 + 0.289i)9-s + (−0.0619 + 0.826i)10-s + (0.241 − 1.05i)11-s + (0.0234 + 0.0597i)12-s + (0.0224 + 0.299i)13-s + (1.04 + 0.157i)14-s + (−0.112 + 0.104i)15-s + (0.256 − 0.123i)16-s + (1.33 + 0.912i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.61781 + 0.905026i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61781 + 0.905026i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-9.87e3 - 7.04e3i)T \) |
| good | 2 | \( 1 + (-2.53 + 3.18i)T + (-7.12 - 31.1i)T^{2} \) |
| 3 | \( 1 + (-2.05 + 0.309i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (53.1 - 36.2i)T + (1.14e3 - 2.90e3i)T^{2} \) |
| 7 | \( 1 + (-95.1 - 164. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-96.9 + 424. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-13.6 - 182. i)T + (-3.67e5 + 5.53e4i)T^{2} \) |
| 17 | \( 1 + (-1.59e3 - 1.08e3i)T + (5.18e5 + 1.32e6i)T^{2} \) |
| 19 | \( 1 + (580. + 179. i)T + (2.04e6 + 1.39e6i)T^{2} \) |
| 23 | \( 1 + (1.03e3 + 963. i)T + (4.80e5 + 6.41e6i)T^{2} \) |
| 29 | \( 1 + (-2.77e3 - 418. i)T + (1.95e7 + 6.04e6i)T^{2} \) |
| 31 | \( 1 + (3.01e3 + 7.68e3i)T + (-2.09e7 + 1.94e7i)T^{2} \) |
| 37 | \( 1 + (1.49e3 - 2.59e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-1.16e4 + 1.46e4i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 47 | \( 1 + (491. + 2.15e3i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.32e3 - 1.76e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-4.14e4 + 1.99e4i)T + (4.45e8 - 5.58e8i)T^{2} \) |
| 61 | \( 1 + (3.30e3 - 8.41e3i)T + (-6.19e8 - 5.74e8i)T^{2} \) |
| 67 | \( 1 + (-6.75e4 - 2.08e4i)T + (1.11e9 + 7.60e8i)T^{2} \) |
| 71 | \( 1 + (-1.05e4 + 9.81e3i)T + (1.34e8 - 1.79e9i)T^{2} \) |
| 73 | \( 1 + (948. + 1.26e4i)T + (-2.04e9 + 3.08e8i)T^{2} \) |
| 79 | \( 1 + (3.62e4 + 6.28e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.03e4 + 1.55e3i)T + (3.76e9 - 1.16e9i)T^{2} \) |
| 89 | \( 1 + (9.30e4 - 1.40e4i)T + (5.33e9 - 1.64e9i)T^{2} \) |
| 97 | \( 1 + (1.97e4 - 8.66e4i)T + (-7.73e9 - 3.72e9i)T^{2} \) |
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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
−14.85172855635933626295577804201, −14.09511630496501269808849237391, −12.43718187608700772968193341996, −11.56647611551443740209597535596, −10.97597717071588990300724061877, −8.572991138402195345866448278576, −7.85675678620355949719345133324, −5.72150872161466285655387746281, −3.76908999687189161766090259574, −2.49170513999636567288912544723,
0.931954902885577058030957567895, 4.06693852077435789213579880358, 5.21873275230291051921486698418, 7.12350238700005288258101525668, 8.079334508444901786160744984923, 9.910772053949539618736910486766, 11.24442512868631694702218617446, 12.45746977301810547715671905640, 14.09593740896203492814427855926, 14.55715885484127098000107320730
