| L(s) = 1 | + (1.75 − 0.183i)2-s + (0.0685 + 0.322i)3-s + (1.07 − 0.228i)4-s + (2.20 − 0.357i)5-s + (0.179 + 0.552i)6-s + (−2.55 − 0.690i)7-s + (−1.51 + 0.490i)8-s + (2.64 − 1.17i)9-s + (3.79 − 1.03i)10-s + (1.99 + 0.890i)11-s + (0.147 + 0.330i)12-s + (−2.54 + 3.50i)13-s + (−4.59 − 0.739i)14-s + (0.266 + 0.687i)15-s + (−4.55 + 2.02i)16-s + (−2.97 − 2.68i)17-s + ⋯ |
| L(s) = 1 | + (1.23 − 0.130i)2-s + (0.0395 + 0.186i)3-s + (0.536 − 0.114i)4-s + (0.987 − 0.159i)5-s + (0.0732 + 0.225i)6-s + (−0.965 − 0.261i)7-s + (−0.534 + 0.173i)8-s + (0.880 − 0.391i)9-s + (1.20 − 0.326i)10-s + (0.602 + 0.268i)11-s + (0.0425 + 0.0954i)12-s + (−0.705 + 0.971i)13-s + (−1.22 − 0.197i)14-s + (0.0688 + 0.177i)15-s + (−1.13 + 0.507i)16-s + (−0.721 − 0.650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07690 - 0.0709920i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07690 - 0.0709920i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.20 + 0.357i)T \) |
| 7 | \( 1 + (2.55 + 0.690i)T \) |
| good | 2 | \( 1 + (-1.75 + 0.183i)T + (1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-0.0685 - 0.322i)T + (-2.74 + 1.22i)T^{2} \) |
| 11 | \( 1 + (-1.99 - 0.890i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (2.54 - 3.50i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.97 + 2.68i)T + (1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (5.05 + 1.07i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.36 + 0.143i)T + (22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-2.10 + 6.47i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.71 - 5.24i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (1.79 + 4.02i)T + (-24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (-7.67 - 5.57i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 0.649iT - 43T^{2} \) |
| 47 | \( 1 + (1.55 - 1.39i)T + (4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-2.44 - 11.5i)T + (-48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.02 + 9.73i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (0.429 + 4.08i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-5.89 - 5.30i)T + (7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (-1.52 + 4.70i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.98 + 13.4i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-9.73 - 10.8i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.66 + 0.539i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.805 - 7.66i)T + (-87.0 + 18.5i)T^{2} \) |
| 97 | \( 1 + (1.06 + 0.345i)T + (78.4 + 57.0i)T^{2} \) |
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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
−12.77504740153446086501071253583, −12.24397376998134669273484929534, −10.83324001752502969539631025516, −9.473925858398877081537342915390, −9.216829379754783158361596112518, −6.84015995219554350052509754294, −6.34194592229052060363199220792, −4.81693096849091548827376314396, −3.99190334177290381327646866143, −2.38460112854451510750782852583,
2.42158474231403071724566716077, 3.84859429196879240210093344464, 5.21883316277218257503493403514, 6.22062427186069805570309438489, 6.96278281425107415085957935983, 8.803362151957284324691143837488, 9.814909572039377495261790823881, 10.73956235008329316379114156409, 12.39505357818652808001856787813, 12.92577263489826746205347991905
