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
−11.90707484947077803661022476571, −10.60041112180449276882142036060, −10.16887202705296213244618485603, −9.168963309731528273613488516193, −7.53454735267595954448268493353, −6.68954454959922752623467270801, −5.91729795142868436442151583428, −4.61042323565496317725993357258, −2.69422626129935347788935221121, −1.58406772837284712867432854903,
1.23234719328615716301731952045, 2.72610661876107847855791528730, 3.98331338814158066898800113068, 5.72499124350208485540869868687, 6.41093945082876176548053428276, 7.80543119141726374006135437130, 8.775423422319367396391637627719, 9.770844239503679050756992180200, 10.94538505149182842878020267072, 11.68962028794559806784252740087