| 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
−14.81034422766351914223316843473, −13.90325113613239043595290210902, −12.45806606804380620332974049735, −11.08858465724211684002187851270, −9.552046162524830831907749028025, −8.519006728396721968237977737924, −7.909674822067458465925749872574, −6.40246319142316297433566783987, −5.39134860204460957453882527704, −3.58453651954180494619760277465,
1.55057608582294596304658829744, 3.26388257276622617303370652926, 4.63919329840883997950289548027, 7.06043399765416591802931334980, 8.387899568225647165852808044877, 9.293431898479525404347380388295, 10.74697822526143374724120072022, 11.28955106454052060713874151971, 12.33579576416025066760311363635, 13.66182093231843542127020899625
