| L(s) = 1 | + (1.92 − 1.39i)2-s + (1.98 − 6.11i)3-s + (−0.729 + 2.24i)4-s + (−10.2 + 4.41i)5-s + (−4.71 − 14.5i)6-s + 25.3·7-s + (7.60 + 23.3i)8-s + (−11.6 − 8.43i)9-s + (−13.5 + 22.8i)10-s + (−19.6 + 14.2i)11-s + (12.2 + 8.92i)12-s + (−67.4 − 49.0i)13-s + (48.6 − 35.3i)14-s + (6.56 + 71.5i)15-s + (31.9 + 23.2i)16-s + (12.7 + 39.3i)17-s + ⋯ |
| L(s) = 1 | + (0.679 − 0.493i)2-s + (0.382 − 1.17i)3-s + (−0.0912 + 0.280i)4-s + (−0.918 + 0.394i)5-s + (−0.321 − 0.987i)6-s + 1.36·7-s + (0.336 + 1.03i)8-s + (−0.429 − 0.312i)9-s + (−0.429 + 0.721i)10-s + (−0.538 + 0.391i)11-s + (0.295 + 0.214i)12-s + (−1.43 − 1.04i)13-s + (0.929 − 0.675i)14-s + (0.113 + 1.23i)15-s + (0.499 + 0.363i)16-s + (0.182 + 0.561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.39657 - 0.716459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39657 - 0.716459i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.2 - 4.41i)T \) |
| good | 2 | \( 1 + (-1.92 + 1.39i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-1.98 + 6.11i)T + (-21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 - 25.3T + 343T^{2} \) |
| 11 | \( 1 + (19.6 - 14.2i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (67.4 + 49.0i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-12.7 - 39.3i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (19.0 + 58.5i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (36.6 - 26.6i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-2.63 + 8.10i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (22.6 + 69.7i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-120. - 87.6i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-133. - 97.0i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (62.2 - 191. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-47.0 + 144. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (659. + 479. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (381. - 277. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-65.2 - 200. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (31.0 - 95.5i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-944. + 686. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (225. - 694. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-255. - 785. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-870. + 632. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-58.4 + 179. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−17.36070890345921576762815753613, −15.13407978862334720092639706436, −14.24489032915357039621248036668, −12.86803696745336430059176613871, −12.12612913961710530201663121308, −10.91542710396395776536060231026, −7.964861615682010864936579145180, −7.64721009249513642286886875862, −4.72193956675949852351654842003, −2.50640875087208887765173888047,
4.24942184566986523247578851266, 5.03241553339331816613948492158, 7.61820190463999420915171221661, 9.233283566602964067293413271690, 10.70078816978957335518940246897, 12.19409418227957506450810901500, 14.20313109512466237460603973712, 14.75901035746001860401134190798, 15.77409106245202966339824031207, 16.68565659417767862409443560172
