| L(s) = 1 | + (0.177 − 0.546i)2-s + (6.20 + 4.50i)4-s + (9.45 − 5.96i)5-s − 2.67·7-s + (7.28 − 5.29i)8-s + (−1.58 − 6.22i)10-s + (19.4 − 59.9i)11-s + (4.38 + 13.4i)13-s + (−0.475 + 1.46i)14-s + (17.3 + 53.4i)16-s + (24.5 − 17.8i)17-s + (−22.2 + 16.1i)19-s + (85.5 + 5.57i)20-s + (−29.3 − 21.2i)22-s + (−35.9 + 110. i)23-s + ⋯ |
| L(s) = 1 | + (0.0627 − 0.193i)2-s + (0.775 + 0.563i)4-s + (0.845 − 0.533i)5-s − 0.144·7-s + (0.321 − 0.233i)8-s + (−0.0500 − 0.196i)10-s + (0.533 − 1.64i)11-s + (0.0934 + 0.287i)13-s + (−0.00907 + 0.0279i)14-s + (0.271 + 0.834i)16-s + (0.350 − 0.254i)17-s + (−0.268 + 0.195i)19-s + (0.956 + 0.0623i)20-s + (−0.283 − 0.206i)22-s + (−0.325 + 1.00i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.52730 - 0.654836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52730 - 0.654836i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-9.45 + 5.96i)T \) |
| good | 2 | \( 1 + (-0.177 + 0.546i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 + 2.67T + 343T^{2} \) |
| 11 | \( 1 + (-19.4 + 59.9i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-4.38 - 13.4i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-24.5 + 17.8i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (22.2 - 16.1i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (35.9 - 110. i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (32.5 + 23.6i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-180. + 130. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-25.6 - 78.8i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (44.5 + 137. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 433.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-371. - 269. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (200. + 145. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (95.9 + 295. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-0.644 + 1.98i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (529. - 384. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (734. + 533. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (271. - 835. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-921. - 669. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (798. - 579. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (305. - 940. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (1.24e3 + 903. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−11.68962028794559806784252740087, −10.94538505149182842878020267072, −9.770844239503679050756992180200, −8.775423422319367396391637627719, −7.80543119141726374006135437130, −6.41093945082876176548053428276, −5.72499124350208485540869868687, −3.98331338814158066898800113068, −2.72610661876107847855791528730, −1.23234719328615716301731952045,
1.58406772837284712867432854903, 2.69422626129935347788935221121, 4.61042323565496317725993357258, 5.91729795142868436442151583428, 6.68954454959922752623467270801, 7.53454735267595954448268493353, 9.168963309731528273613488516193, 10.16887202705296213244618485603, 10.60041112180449276882142036060, 11.90707484947077803661022476571
