| L(s) = 1 | + (−6.12 + 4.44i)2-s + (−7.99 + 24.5i)3-s + (7.81 − 24.0i)4-s + (−12.4 + 54.4i)5-s + (−60.4 − 186. i)6-s + 130.·7-s + (−15.7 − 48.3i)8-s + (−344. − 250. i)9-s + (−166. − 389. i)10-s + (−118. + 86.3i)11-s + (528. + 384. i)12-s + (623. + 452. i)13-s + (−801. + 582. i)14-s + (−1.24e3 − 742. i)15-s + (965. + 701. i)16-s + (184. + 566. i)17-s + ⋯ |
| L(s) = 1 | + (−1.08 + 0.786i)2-s + (−0.512 + 1.57i)3-s + (0.244 − 0.751i)4-s + (−0.223 + 0.974i)5-s + (−0.685 − 2.11i)6-s + 1.00·7-s + (−0.0868 − 0.267i)8-s + (−1.41 − 1.03i)9-s + (−0.524 − 1.23i)10-s + (−0.296 + 0.215i)11-s + (1.06 + 0.770i)12-s + (1.02 + 0.743i)13-s + (−1.09 + 0.794i)14-s + (−1.42 − 0.851i)15-s + (0.943 + 0.685i)16-s + (0.154 + 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.255066 - 0.507080i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.255066 - 0.507080i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (12.4 - 54.4i)T \) |
| good | 2 | \( 1 + (6.12 - 4.44i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (7.99 - 24.5i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 - 130.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (118. - 86.3i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-623. - 452. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-184. - 566. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (520. + 1.60e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (3.11e3 - 2.26e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-2.01e3 + 6.21e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-876. - 2.69e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (6.40e3 + 4.65e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.46e4 - 1.06e4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (5.52e3 - 1.70e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (3.01e3 - 9.26e3i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (5.42e3 + 3.94e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-4.34e3 + 3.15e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.24e3 - 3.82e3i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.18e3 - 6.72e3i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.95e4 + 2.14e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.50e4 - 7.71e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (579. + 1.78e3i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-7.42e4 + 5.39e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (1.76e4 - 5.44e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−17.39351920211077852804013748003, −15.94582300877344114220994327248, −15.50006691514434260980066881882, −14.31080949163350897499901915216, −11.42113811746601814790448496275, −10.57310019098337877010076313404, −9.382065029648859906770817886765, −7.955444654340036492914501085416, −6.14975393623049227574368850342, −4.11120777845189937476914212892,
0.61226734745617300490486632224, 1.73864421161478941714985463709, 5.62698378597614788308095219257, 7.905225528934991528074750114406, 8.505322799824196644835020343867, 10.67445104285795955742004478611, 11.81992587402633173024147203902, 12.63213983148468275274432341595, 14.07248045599983621317720763036, 16.38043321244949789291215725407
