| L(s) = 1 | + (−0.195 − 0.338i)2-s + (5.35 − 14.6i)3-s + (15.9 − 27.5i)4-s + (48.2 − 83.6i)5-s + (−5.99 + 1.04i)6-s + (24.5 + 42.4i)7-s − 24.9·8-s + (−185. − 156. i)9-s − 37.6·10-s + (288. + 499. i)11-s + (−318. − 380. i)12-s + (−288. + 499. i)13-s + (9.56 − 16.5i)14-s + (−965. − 1.15e3i)15-s + (−504. − 874. i)16-s + 346.·17-s + ⋯ |
| L(s) = 1 | + (−0.0345 − 0.0597i)2-s + (0.343 − 0.939i)3-s + (0.497 − 0.861i)4-s + (0.863 − 1.49i)5-s + (−0.0679 + 0.0118i)6-s + (0.188 + 0.327i)7-s − 0.137·8-s + (−0.763 − 0.645i)9-s − 0.119·10-s + (0.718 + 1.24i)11-s + (−0.638 − 0.763i)12-s + (−0.473 + 0.819i)13-s + (0.0130 − 0.0225i)14-s + (−1.10 − 1.32i)15-s + (−0.492 − 0.853i)16-s + 0.290·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.19023 - 2.06924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19023 - 2.06924i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.35 + 14.6i)T \) |
| 7 | \( 1 + (-24.5 - 42.4i)T \) |
| good | 2 | \( 1 + (0.195 + 0.338i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-48.2 + 83.6i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-288. - 499. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (288. - 499. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 346.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.30e3 - 2.25e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-740. - 1.28e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.90e3 + 3.29e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 4.47e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (834. - 1.44e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.61e3 + 2.79e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.42e4 - 2.46e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 5.61e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (9.25e3 - 1.60e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.37e4 + 4.10e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.91e4 + 5.05e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 1.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (104. + 181. i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (5.59e4 + 9.69e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 5.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.18e4 - 5.51e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.70256841664895534181590074943, −12.33159272581903804480964312451, −11.81330365484290254074598298176, −9.635225987674153251097028461619, −9.216385497429223973836805371372, −7.49213480870342822966764832280, −6.12461283964874378786720859194, −4.97259480126747466796940758082, −1.99583303629872568179703853640, −1.22418454852401107665346515863,
2.71030184055307122944338793430, 3.54582851332736552974791890855, 5.78592680193553314304429165501, 7.13713937505213149357798141169, 8.462200578781341151579467972744, 9.966773997101043374387376133001, 10.78328892358665116741352115451, 11.77053384855958558599916983992, 13.69816224344652264741273715581, 14.28196827539004429946568667822
