| L(s) = 1 | + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (1.55 + 0.565i)5-s + (0.766 + 0.642i)6-s + (−0.0923 + 0.160i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.286 + 1.62i)10-s + (2.17 + 3.76i)11-s + (−0.499 + 0.866i)12-s + (−4.96 − 4.16i)13-s + (−0.173 − 0.0632i)14-s + (1.55 − 0.565i)15-s + (0.766 − 0.642i)16-s + (−0.368 − 2.08i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.442 − 0.371i)3-s + (−0.469 + 0.171i)4-s + (0.694 + 0.252i)5-s + (0.312 + 0.262i)6-s + (−0.0349 + 0.0604i)7-s + (−0.176 − 0.306i)8-s + (0.0578 − 0.328i)9-s + (−0.0907 + 0.514i)10-s + (0.655 + 1.13i)11-s + (−0.144 + 0.249i)12-s + (−1.37 − 1.15i)13-s + (−0.0464 − 0.0168i)14-s + (0.400 − 0.145i)15-s + (0.191 − 0.160i)16-s + (−0.0893 − 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18432 + 0.431935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18432 + 0.431935i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (4.11 - 1.43i)T \) |
| good | 5 | \( 1 + (-1.55 - 0.565i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0923 - 0.160i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.17 - 3.76i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.96 + 4.16i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.368 + 2.08i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.0996 - 0.0362i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.692 + 3.92i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.61 + 2.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + (6.61 - 5.55i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.0393 + 0.0143i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.37 - 7.80i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-8.65 + 3.15i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.75 - 9.93i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (3.37 - 1.22i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.38 - 7.86i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.79 - 1.38i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 9.76i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.49 + 7.96i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (4.22 - 7.32i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (13.7 + 11.5i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.85 - 10.5i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.83328527540605487694555401660, −12.85396169918379882960344650446, −12.02547787011539420628713101244, −10.12789553089038681578424363425, −9.488079619879561479425194328134, −8.051376573390140655425676136383, −7.09585023304894204786799674667, −5.98635482940148055999115258791, −4.50831837217890108253404145998, −2.47686776152535551816621986439,
2.11885032600494272846516133560, 3.81164061229600247761355784863, 5.15577027918857297465035590924, 6.67951729380984350864459846999, 8.539648537794574398870343757038, 9.303875131969897676117058457900, 10.25391231050297277537133845960, 11.37608284252263915134716748086, 12.44005328843581978233366437321, 13.60372680693068017613248868534
