| L(s) = 1 | + (3.32 − 3.08i)2-s + (18.9 − 2.85i)3-s + (−0.858 + 11.4i)4-s + (20.6 − 52.7i)5-s + (54.1 − 67.8i)6-s + (−50.0 − 119. i)7-s + (122. + 153. i)8-s + (118. − 36.6i)9-s + (−93.7 − 238. i)10-s + (535. + 165. i)11-s + (16.4 + 219. i)12-s + (−40.3 − 176. i)13-s + (−534. − 242. i)14-s + (241. − 1.05e3i)15-s + (518. + 78.1i)16-s + (−1.25e3 + 854. i)17-s + ⋯ |
| L(s) = 1 | + (0.586 − 0.544i)2-s + (1.21 − 0.183i)3-s + (−0.0268 + 0.357i)4-s + (0.370 − 0.943i)5-s + (0.613 − 0.769i)6-s + (−0.386 − 0.922i)7-s + (0.678 + 0.850i)8-s + (0.488 − 0.150i)9-s + (−0.296 − 0.755i)10-s + (1.33 + 0.411i)11-s + (0.0329 + 0.439i)12-s + (−0.0662 − 0.290i)13-s + (−0.728 − 0.331i)14-s + (0.277 − 1.21i)15-s + (0.506 + 0.0763i)16-s + (−1.05 + 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.90355 - 1.51130i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.90355 - 1.51130i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (50.0 + 119. i)T \) |
| good | 2 | \( 1 + (-3.32 + 3.08i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (-18.9 + 2.85i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (-20.6 + 52.7i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (-535. - 165. i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (40.3 + 176. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (1.25e3 - 854. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (819. + 1.42e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.94e3 - 1.32e3i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (2.26e3 + 1.09e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (3.20e3 - 5.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-1.03e3 - 1.38e4i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (3.16e3 + 3.97e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-3.80e3 + 4.77e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-3.99e3 + 3.71e3i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (1.02e3 - 1.37e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-2.41e3 - 6.16e3i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (3.98e3 + 5.31e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (-1.02e4 + 1.78e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-5.26e4 + 2.53e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-1.19e4 - 1.10e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (3.94e4 + 6.82e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-8.96e3 + 3.92e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (1.13e5 - 3.49e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 + 8.09e4T + 8.58e9T^{2} \) |
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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
−13.97110922446472151378161228289, −13.28010164506263236375054536718, −12.59327794609765760167368670556, −11.06162415641003785139988442861, −9.312127470152680140292145111886, −8.498867617961927691601521412770, −7.02220396888275502321618847750, −4.58472670888538098893068716189, −3.41662438348605470162765144133, −1.70572807883937163137011771797,
2.34759541602237640513419540669, 3.90277454439752144930670643394, 5.95207337648768383604309881905, 6.93795589705997544704239197638, 8.860104608631459135151131151386, 9.645111057213210976850942401626, 11.19814975634717849082116354917, 12.97311784730305397207111826883, 14.21765665422438056468262434860, 14.54662303718410541262267110303
