| L(s) = 1 | + (−2.82 + 15.7i)2-s + (30.1 + 9.80i)3-s + (−240. − 88.9i)4-s + (134. + 610. i)5-s + (−239. + 447. i)6-s − 4.00e3i·7-s + (2.07e3 − 3.52e3i)8-s + (−4.49e3 − 3.26e3i)9-s + (−9.99e3 + 397. i)10-s + (1.13e4 + 1.56e4i)11-s + (−6.37e3 − 5.03e3i)12-s + (−1.64e4 − 1.19e4i)13-s + (6.30e4 + 1.13e4i)14-s + (−1.92e3 + 1.97e4i)15-s + (4.97e4 + 4.26e4i)16-s + (2.09e4 + 6.45e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.176 + 0.984i)2-s + (0.372 + 0.121i)3-s + (−0.937 − 0.347i)4-s + (0.215 + 0.976i)5-s + (−0.184 + 0.345i)6-s − 1.66i·7-s + (0.507 − 0.861i)8-s + (−0.684 − 0.497i)9-s + (−0.999 + 0.0397i)10-s + (0.775 + 1.06i)11-s + (−0.307 − 0.242i)12-s + (−0.576 − 0.418i)13-s + (1.64 + 0.294i)14-s + (−0.0379 + 0.389i)15-s + (0.758 + 0.651i)16-s + (0.251 + 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.868605 + 1.49721i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.868605 + 1.49721i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 - 15.7i)T \) |
| 5 | \( 1 + (-134. - 610. i)T \) |
| good | 3 | \( 1 + (-30.1 - 9.80i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 4.00e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.13e4 - 1.56e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (1.64e4 + 1.19e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-2.09e4 - 6.45e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-7.38e4 + 2.40e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-2.51e5 - 3.46e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (-1.11e5 + 3.41e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.97e5 + 6.42e4i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-2.39e6 - 1.73e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-3.15e6 - 2.29e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 2.21e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (3.71e6 + 1.20e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-1.33e5 + 4.11e5i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (6.06e6 - 8.35e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (5.78e6 - 4.20e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (2.12e7 - 6.89e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-4.44e7 - 1.44e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (9.05e6 - 6.57e6i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-2.61e7 - 8.50e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (5.64e7 - 1.83e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-4.55e7 + 3.31e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-4.54e7 + 1.39e8i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−13.08171571362536581211552611048, −11.35723015220858562036702498398, −10.03549862787283846672241199042, −9.544189387759257345653172138600, −7.82409797865676832116249520794, −7.15005825166004976219633076345, −6.14225910486812411988830209345, −4.42652456215709410200389987684, −3.30914237236851163229948842588, −1.07781668079781523203108416136,
0.60413129863197362269833982653, 2.08484795630789368994477797319, 3.02952592893247170968229984680, 4.84101874230968082416460475324, 5.77272846990860824999982819172, 8.082604977352170564280910929983, 9.027398701533829272009090760456, 9.271748383595184566153289840443, 11.12431670417426265333866318399, 11.97740055482859243084705289858
