| L(s) = 1 | + (−3.62 − 4.98i)2-s + (10.7 + 3.49i)3-s + (−1.84 + 5.68i)4-s + (55.8 − 2.36i)5-s + (−21.5 − 66.1i)6-s − 133. i·7-s + (−152. + 49.5i)8-s + (−93.3 − 67.8i)9-s + (−214. − 269. i)10-s + (386. − 280. i)11-s + (−39.6 + 54.6i)12-s + (−3.32 + 4.57i)13-s + (−663. + 481. i)14-s + (608. + 169. i)15-s + (954. + 693. i)16-s + (72.0 − 23.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.640 − 0.881i)2-s + (0.689 + 0.223i)3-s + (−0.0577 + 0.177i)4-s + (0.999 − 0.0422i)5-s + (−0.243 − 0.750i)6-s − 1.02i·7-s + (−0.842 + 0.273i)8-s + (−0.384 − 0.279i)9-s + (−0.676 − 0.853i)10-s + (0.961 − 0.698i)11-s + (−0.0795 + 0.109i)12-s + (−0.00545 + 0.00750i)13-s + (−0.904 + 0.656i)14-s + (0.697 + 0.194i)15-s + (0.931 + 0.677i)16-s + (0.0605 − 0.0196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.968381 - 1.07473i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968381 - 1.07473i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-55.8 + 2.36i)T \) |
| good | 2 | \( 1 + (3.62 + 4.98i)T + (-9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (-10.7 - 3.49i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 + 133. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-386. + 280. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (3.32 - 4.57i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-72.0 + 23.4i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-971. - 2.99e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.12e3 + 1.54e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (563. - 1.73e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-2.44e3 - 7.51e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (4.49e3 - 6.18e3i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-3.57e3 - 2.59e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 6.35e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.72e3 - 884. i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-7.31e3 - 2.37e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.76e4 + 2.00e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.43e4 + 1.03e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.89e4 - 6.15e3i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-8.88e3 + 2.73e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (4.58e4 + 6.31e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.56e4 - 7.89e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (5.99e4 - 1.94e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-9.85e4 + 7.15e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.58e5 - 5.14e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.57037192847998376356138990132, −14.46441234360567249083967346860, −13.97769361978816541123107752678, −12.08687632446406212318501559914, −10.56167741903810816832005207723, −9.664403859118656641551440793872, −8.527158392949537155920649006327, −6.14847482470126728898458099740, −3.34159082789945281198064696956, −1.32781170337533916592048788027,
2.48776685811202860222952009910, 5.76787178571777114953764943692, 7.23945715810925429979723780654, 8.838836469901380659143911937088, 9.440166084605228180314005607381, 11.80336813982835433698519181147, 13.36045152892043679505457016490, 14.63191522874877960709502022061, 15.59748607737644160367288480707, 17.12802340451986585368858523923
