| L(s) = 1 | + (−0.385 − 2.18i)2-s + (0.794 − 0.666i)3-s + (−2.76 + 1.00i)4-s + (−0.939 − 0.342i)5-s + (−1.76 − 1.48i)6-s + (1.01 − 1.75i)7-s + (1.04 + 1.81i)8-s + (−0.334 + 1.89i)9-s + (−0.385 + 2.18i)10-s + (−0.0424 − 0.0734i)11-s + (−1.52 + 2.64i)12-s + (4.38 + 3.67i)13-s + (−4.22 − 1.53i)14-s + (−0.974 + 0.354i)15-s + (−0.945 + 0.793i)16-s + (−0.439 − 2.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.272 − 1.54i)2-s + (0.458 − 0.384i)3-s + (−1.38 + 0.502i)4-s + (−0.420 − 0.152i)5-s + (−0.720 − 0.604i)6-s + (0.382 − 0.662i)7-s + (0.369 + 0.640i)8-s + (−0.111 + 0.631i)9-s + (−0.122 + 0.692i)10-s + (−0.0127 − 0.0221i)11-s + (−0.440 + 0.762i)12-s + (1.21 + 1.01i)13-s + (−1.13 − 0.411i)14-s + (−0.251 + 0.0915i)15-s + (−0.236 + 0.198i)16-s + (−0.106 − 0.604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.343416 - 0.863672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.343416 - 0.863672i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-3.21 + 2.94i)T \) |
| good | 2 | \( 1 + (0.385 + 2.18i)T + (-1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.794 + 0.666i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.01 + 1.75i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0424 + 0.0734i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.38 - 3.67i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.439 + 2.49i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.290 + 0.105i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.455 + 2.58i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (4.03 - 6.99i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 + (4.50 - 3.77i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.611 + 0.222i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.19 + 6.79i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (13.6 - 4.97i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.29 - 7.35i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (12.5 - 4.55i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.54 + 8.74i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-13.4 - 4.90i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.31 + 6.97i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-0.0887 + 0.0744i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.48 + 2.57i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.52 + 7.15i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.0392 + 0.222i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.66182093231843542127020899625, −12.33579576416025066760311363635, −11.28955106454052060713874151971, −10.74697822526143374724120072022, −9.293431898479525404347380388295, −8.387899568225647165852808044877, −7.06043399765416591802931334980, −4.63919329840883997950289548027, −3.26388257276622617303370652926, −1.55057608582294596304658829744,
3.58453651954180494619760277465, 5.39134860204460957453882527704, 6.40246319142316297433566783987, 7.909674822067458465925749872574, 8.519006728396721968237977737924, 9.552046162524830831907749028025, 11.08858465724211684002187851270, 12.45806606804380620332974049735, 13.90325113613239043595290210902, 14.81034422766351914223316843473
