| L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−2.12 − 0.697i)5-s + (0.809 + 0.587i)6-s − 4.25·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.00655 + 2.23i)10-s + (−1.51 − 4.64i)11-s + (0.309 − 0.951i)12-s + (−1.43 + 4.41i)13-s + (1.31 + 4.04i)14-s + (2.12 − 0.684i)15-s + (0.309 − 0.951i)16-s + (0.815 + 0.592i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.950 − 0.311i)5-s + (0.330 + 0.239i)6-s − 1.60·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.00207 + 0.707i)10-s + (−0.455 − 1.40i)11-s + (0.0892 − 0.274i)12-s + (−0.397 + 1.22i)13-s + (0.351 + 1.08i)14-s + (0.549 − 0.176i)15-s + (0.0772 − 0.237i)16-s + (0.197 + 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0155833 + 0.120511i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0155833 + 0.120511i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (2.12 + 0.697i)T \) |
| good | 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 + (1.51 + 4.64i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.43 - 4.41i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.815 - 0.592i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1 - 0.726i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.38 + 4.25i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.42 - 2.48i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.826 + 0.600i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.31 + 10.1i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.42 + 4.37i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + (-1.63 + 1.19i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (8.94 - 6.49i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.656 - 2.02i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.19 - 3.68i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (3.03 + 2.20i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (10.1 - 7.37i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.05 - 3.26i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (5.16 - 3.75i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.70 + 4.14i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.693 - 2.13i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.49 + 4.71i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.35197723116990013678804327734, −11.45587392234815584568446224878, −10.57456971507735571064990910840, −9.462287450953543365787961374398, −8.629911242685629168307768750569, −7.14737537220537566714983850100, −5.81986639519649156966693179357, −4.17813204650347352067016675566, −3.17168820940116161791242581814, −0.12975565448505981835450572316,
3.21643557336723267983573575044, 4.90525686306065385207265206139, 6.28335790982375308900983032028, 7.24646523512585555169545395311, 7.931867889108133038822271548954, 9.666726288904029199658937839545, 10.23483516602632224833834479116, 11.71748607743631580042553945873, 12.71222754954233316458842911208, 13.26887751539126525759213998348
