| L(s) = 1 | + (0.637 − 0.463i)2-s + (14.0 − 43.1i)3-s + (−39.3 + 121. i)4-s + (−194. − 200. i)5-s + (−11.0 − 33.9i)6-s − 652.·7-s + (62.1 + 191. i)8-s + (106. + 77.4i)9-s + (−216. − 38.1i)10-s + (−6.15e3 + 4.47e3i)11-s + (4.67e3 + 3.39e3i)12-s + (−6.69e3 − 4.86e3i)13-s + (−416. + 302. i)14-s + (−1.13e4 + 5.56e3i)15-s + (−1.30e4 − 9.49e3i)16-s + (−9.40e3 − 2.89e4i)17-s + ⋯ |
| L(s) = 1 | + (0.0563 − 0.0409i)2-s + (0.299 − 0.921i)3-s + (−0.307 + 0.946i)4-s + (−0.695 − 0.718i)5-s + (−0.0208 − 0.0642i)6-s − 0.719·7-s + (0.0429 + 0.132i)8-s + (0.0487 + 0.0354i)9-s + (−0.0686 − 0.0120i)10-s + (−1.39 + 1.01i)11-s + (0.780 + 0.567i)12-s + (−0.844 − 0.613i)13-s + (−0.0405 + 0.0294i)14-s + (−0.871 + 0.425i)15-s + (−0.797 − 0.579i)16-s + (−0.464 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0187877 + 0.168399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0187877 + 0.168399i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (194. + 200. i)T \) |
| good | 2 | \( 1 + (-0.637 + 0.463i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (-14.0 + 43.1i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 + 652.T + 8.23e5T^{2} \) |
| 11 | \( 1 + (6.15e3 - 4.47e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (6.69e3 + 4.86e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (9.40e3 + 2.89e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-1.54e4 - 4.74e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-2.51e4 + 1.82e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-6.43e4 + 1.98e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (3.38e4 + 1.04e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (2.35e4 + 1.71e4i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-1.40e5 - 1.02e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 4.83e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (5.12e4 - 1.57e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (4.43e5 - 1.36e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.80e5 + 2.03e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (1.29e6 - 9.40e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (6.91e5 + 2.12e6i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-2.51e4 + 7.73e4i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (2.43e6 - 1.77e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-8.05e5 + 2.47e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-1.86e6 - 5.75e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (2.29e6 - 1.66e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (2.05e6 - 6.31e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−15.55470477180659892586714116306, −13.52699609531926559995232566495, −12.71758118602715683011043226496, −12.08982764860304447521254678342, −9.741856868937553867751591890395, −7.940826591149264015667916069008, −7.43072500037159119173461882901, −4.72522525056268286917691502777, −2.69811096085381043265129929329, −0.079644042797257897421867777408,
3.19593161369180295617390868161, 4.88825849976126275113491612823, 6.77399941385077633274341785966, 8.868939575263244417292304912303, 10.19838860005557352557831705316, 10.98057614851268685595435971295, 13.09407417715078924396414050750, 14.50310635864778575812770091281, 15.45283782969855449026013330293, 16.07072347258058711202470214261
