| L(s) = 1 | + (0.422 − 0.906i)2-s + (−1.13 + 1.31i)3-s + (−0.642 − 0.766i)4-s + (−1.05 + 1.97i)5-s + (0.710 + 1.57i)6-s + (−0.0450 + 0.168i)7-s + (−0.965 + 0.258i)8-s + (−0.438 − 2.96i)9-s + (1.34 + 1.78i)10-s + (−1.02 − 0.592i)11-s + (1.73 + 0.0240i)12-s + (−0.862 − 0.604i)13-s + (0.133 + 0.111i)14-s + (−1.39 − 3.61i)15-s + (−0.173 + 0.984i)16-s + (0.00890 − 0.0191i)17-s + ⋯ |
| L(s) = 1 | + (0.298 − 0.640i)2-s + (−0.653 + 0.757i)3-s + (−0.321 − 0.383i)4-s + (−0.471 + 0.881i)5-s + (0.289 + 0.644i)6-s + (−0.0170 + 0.0636i)7-s + (−0.341 + 0.0915i)8-s + (−0.146 − 0.989i)9-s + (0.424 + 0.565i)10-s + (−0.309 − 0.178i)11-s + (0.499 + 0.00693i)12-s + (−0.239 − 0.167i)13-s + (0.0356 + 0.0299i)14-s + (−0.359 − 0.933i)15-s + (−0.0434 + 0.246i)16-s + (0.00216 − 0.00463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00301103 - 0.0569645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00301103 - 0.0569645i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.422 + 0.906i)T \) |
| 3 | \( 1 + (1.13 - 1.31i)T \) |
| 5 | \( 1 + (1.05 - 1.97i)T \) |
| 19 | \( 1 + (3.61 + 2.43i)T \) |
| good | 7 | \( 1 + (0.0450 - 0.168i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.02 + 0.592i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.862 + 0.604i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.00890 + 0.0191i)T + (-10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (5.33 + 0.467i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (6.37 + 2.31i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.85 + 4.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.29 - 2.29i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.39 + 0.245i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (8.71 - 0.762i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.72 + 1.73i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.422 - 0.0369i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (6.34 - 2.30i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (3.04 - 2.55i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.76 - 14.5i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-3.95 + 4.71i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-3.74 - 5.35i)T + (-24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (10.5 + 1.85i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.55 + 5.79i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (0.687 + 3.89i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 4.99i)T + (62.3 + 74.3i)T^{2} \) |
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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
−10.41640837001963492462327986533, −9.872336335854964534600169534641, −8.785621239613671156352260581385, −7.58995998190679371406758712148, −6.40717455542825237419914861623, −5.60561900579637523932065091084, −4.40100252006427046381516605173, −3.64283930648078871783126329558, −2.44485805079431296679559782178, −0.03034544068588882190284538446,
1.85112221010714570835014686832, 3.81470906997553788611324135469, 4.89949927251902220429253404309, 5.64341800625967675182207564407, 6.64034042452825275639582840919, 7.60312284993446062903625963533, 8.177628333261736685735694385926, 9.160756009477164370327781236954, 10.38617497696002325735412555621, 11.40084347028796409239793151497
