| L(s) = 1 | + (0.354 + 0.935i)2-s + (1.62 − 0.853i)3-s + (−0.748 + 0.663i)4-s + (−0.272 − 2.24i)5-s + (1.37 + 1.21i)6-s + (2.00 − 1.05i)7-s + (−0.885 − 0.464i)8-s + (0.213 − 0.308i)9-s + (2.00 − 1.05i)10-s + (−0.379 + 3.12i)11-s + (−0.651 + 1.71i)12-s + (−0.973 + 0.862i)13-s + (1.69 + 1.50i)14-s + (−2.35 − 3.41i)15-s + (0.120 − 0.992i)16-s + (−3.61 + 3.20i)17-s + ⋯ |
| L(s) = 1 | + (0.250 + 0.661i)2-s + (0.939 − 0.492i)3-s + (−0.374 + 0.331i)4-s + (−0.121 − 1.00i)5-s + (0.561 + 0.497i)6-s + (0.759 − 0.398i)7-s + (−0.313 − 0.164i)8-s + (0.0710 − 0.102i)9-s + (0.632 − 0.332i)10-s + (−0.114 + 0.942i)11-s + (−0.188 + 0.495i)12-s + (−0.269 + 0.239i)13-s + (0.453 + 0.402i)14-s + (−0.609 − 0.882i)15-s + (0.0301 − 0.248i)16-s + (−0.877 + 0.777i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57056 + 0.124033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57056 + 0.124033i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.354 - 0.935i)T \) |
| 79 | \( 1 + (-8.88 - 0.183i)T \) |
| good | 3 | \( 1 + (-1.62 + 0.853i)T + (1.70 - 2.46i)T^{2} \) |
| 5 | \( 1 + (0.272 + 2.24i)T + (-4.85 + 1.19i)T^{2} \) |
| 7 | \( 1 + (-2.00 + 1.05i)T + (3.97 - 5.76i)T^{2} \) |
| 11 | \( 1 + (0.379 - 3.12i)T + (-10.6 - 2.63i)T^{2} \) |
| 13 | \( 1 + (0.973 - 0.862i)T + (1.56 - 12.9i)T^{2} \) |
| 17 | \( 1 + (3.61 - 3.20i)T + (2.04 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.20 + 0.544i)T + (16.8 + 8.82i)T^{2} \) |
| 23 | \( 1 - 0.445T + 23T^{2} \) |
| 29 | \( 1 + (0.847 + 1.22i)T + (-10.2 + 27.1i)T^{2} \) |
| 31 | \( 1 + (0.812 + 2.14i)T + (-23.2 + 20.5i)T^{2} \) |
| 37 | \( 1 + (-6.67 - 1.64i)T + (32.7 + 17.1i)T^{2} \) |
| 41 | \( 1 + (0.936 + 7.70i)T + (-39.8 + 9.81i)T^{2} \) |
| 43 | \( 1 + (1.02 + 8.44i)T + (-41.7 + 10.2i)T^{2} \) |
| 47 | \( 1 + (12.7 - 3.14i)T + (41.6 - 21.8i)T^{2} \) |
| 53 | \( 1 + (-6.32 - 3.31i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (9.91 + 8.78i)T + (7.11 + 58.5i)T^{2} \) |
| 61 | \( 1 + (-5.57 - 1.37i)T + (54.0 + 28.3i)T^{2} \) |
| 67 | \( 1 + (-2.38 + 6.29i)T + (-50.1 - 44.4i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 5.98i)T + (40.3 + 58.4i)T^{2} \) |
| 73 | \( 1 + (-6.53 - 5.78i)T + (8.79 + 72.4i)T^{2} \) |
| 83 | \( 1 + (-0.329 + 0.291i)T + (10.0 - 82.3i)T^{2} \) |
| 89 | \( 1 + (15.2 - 7.98i)T + (50.5 - 73.2i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 2.88i)T + (85.8 + 45.0i)T^{2} \) |
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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
−13.05370535617488540371633324811, −12.44535745913479209916726934571, −11.02343528969302351208382974849, −9.455444086182741153517872072478, −8.475101779863978701463215517233, −7.88744382074282021384480151272, −6.80977499845791919249425376796, −5.08125228065014286127327600037, −4.16279407962407723956045868605, −2.01617843353926054112932070991,
2.51284730249004065897902097677, 3.39554497405635312563302666895, 4.83019506682982583785611010978, 6.40100098827492005781103046980, 8.001677684885729707253696536738, 8.880749246102027744945771600118, 9.913975211008272471964896787821, 11.06569844436408624520865655131, 11.53975982650859485606639181255, 13.04181000720760667560526676939
