| L(s) = 1 | + (1.14 + 1.36i)2-s + (−0.815 + 2.24i)3-s + (−0.205 + 1.16i)4-s + (1.67 − 1.47i)5-s + (−3.99 + 1.45i)6-s + (−3.67 − 2.11i)7-s + (1.25 − 0.727i)8-s + (−2.05 − 1.72i)9-s + (3.94 + 0.596i)10-s + (0.245 + 0.425i)11-s + (−2.44 − 1.41i)12-s + (1.42 + 3.91i)13-s + (−1.31 − 7.45i)14-s + (1.94 + 4.96i)15-s + (4.66 + 1.69i)16-s + (−1.30 − 1.55i)17-s + ⋯ |
| L(s) = 1 | + (0.811 + 0.966i)2-s + (−0.470 + 1.29i)3-s + (−0.102 + 0.583i)4-s + (0.749 − 0.661i)5-s + (−1.63 + 0.594i)6-s + (−1.38 − 0.801i)7-s + (0.445 − 0.257i)8-s + (−0.684 − 0.574i)9-s + (1.24 + 0.188i)10-s + (0.0740 + 0.128i)11-s + (−0.706 − 0.407i)12-s + (0.394 + 1.08i)13-s + (−0.351 − 1.99i)14-s + (0.502 + 1.28i)15-s + (1.16 + 0.424i)16-s + (−0.317 − 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.874460 + 0.969814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874460 + 0.969814i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.67 + 1.47i)T \) |
| 19 | \( 1 + (1.86 + 3.93i)T \) |
| good | 2 | \( 1 + (-1.14 - 1.36i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (0.815 - 2.24i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (3.67 + 2.11i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.245 - 0.425i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.42 - 3.91i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.30 + 1.55i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (4.32 + 0.763i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.49 - 2.09i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.04 - 3.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.14iT - 37T^{2} \) |
| 41 | \( 1 + (4.10 + 1.49i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-10.4 + 1.84i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.68 - 2.00i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (11.2 + 1.98i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.415 + 0.348i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.36 - 13.4i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.60 + 5.48i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.04 - 5.94i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.702 - 1.93i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.01 - 1.82i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.02 - 4.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.07 - 1.48i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.62 - 4.32i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−14.21878173477181487837175537131, −13.53760832386269962282275794990, −12.57040021985648464772362005886, −10.80563975227422496658553709694, −9.905732088177710887935687779973, −9.087157370479048842113542507092, −6.90621923433615813567999447604, −6.05740107260613164943776203805, −4.79672287777540983686233826028, −3.96001203404129427417843609768,
2.09992894173265906339158550227, 3.34258793214831320761889058589, 5.82309960551606939698443631941, 6.33499859391224438286273850076, 7.914216229058808494880918619099, 9.775146862308427494519891464919, 10.80543165234560867653470584644, 11.99423621825174813376697583793, 12.78004276626084827136486557740, 13.17983440894532077829115333498
