| L(s) = 1 | + (0.540 − 0.410i)2-s + (−1.38 + 2.61i)3-s + (−0.411 + 1.48i)4-s + (2.62 − 2.48i)5-s + (0.325 + 1.98i)6-s + (−1.68 − 1.98i)7-s + (0.889 + 2.23i)8-s + (−3.23 − 4.76i)9-s + (0.396 − 2.41i)10-s + (1.31 − 0.788i)11-s + (−3.30 − 3.13i)12-s + (1.04 − 1.53i)13-s + (−1.72 − 0.379i)14-s + (2.85 + 10.2i)15-s + (−1.24 − 0.746i)16-s + (−2.29 + 2.69i)17-s + ⋯ |
| L(s) = 1 | + (0.382 − 0.290i)2-s + (−0.800 + 1.50i)3-s + (−0.205 + 0.741i)4-s + (1.17 − 1.11i)5-s + (0.132 + 0.809i)6-s + (−0.635 − 0.748i)7-s + (0.314 + 0.788i)8-s + (−1.07 − 1.58i)9-s + (0.125 − 0.764i)10-s + (0.395 − 0.237i)11-s + (−0.955 − 0.904i)12-s + (0.288 − 0.425i)13-s + (−0.460 − 0.101i)14-s + (0.738 + 2.65i)15-s + (−0.310 − 0.186i)16-s + (−0.555 + 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.834536 + 0.308295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.834536 + 0.308295i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-7.17 + 2.73i)T \) |
| good | 2 | \( 1 + (-0.540 + 0.410i)T + (0.535 - 1.92i)T^{2} \) |
| 3 | \( 1 + (1.38 - 2.61i)T + (-1.68 - 2.48i)T^{2} \) |
| 5 | \( 1 + (-2.62 + 2.48i)T + (0.270 - 4.99i)T^{2} \) |
| 7 | \( 1 + (1.68 + 1.98i)T + (-1.13 + 6.90i)T^{2} \) |
| 11 | \( 1 + (-1.31 + 0.788i)T + (5.15 - 9.71i)T^{2} \) |
| 13 | \( 1 + (-1.04 + 1.53i)T + (-4.81 - 12.0i)T^{2} \) |
| 17 | \( 1 + (2.29 - 2.69i)T + (-2.75 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.245 - 0.113i)T + (12.3 + 14.4i)T^{2} \) |
| 23 | \( 1 + (3.08 - 1.03i)T + (18.3 - 13.9i)T^{2} \) |
| 29 | \( 1 + (0.794 + 0.604i)T + (7.75 + 27.9i)T^{2} \) |
| 31 | \( 1 + (-2.26 + 1.04i)T + (20.0 - 23.6i)T^{2} \) |
| 37 | \( 1 + (3.21 - 8.06i)T + (-26.8 - 25.4i)T^{2} \) |
| 41 | \( 1 + (-4.43 - 1.49i)T + (32.6 + 24.8i)T^{2} \) |
| 43 | \( 1 + (10.2 + 6.17i)T + (20.1 + 37.9i)T^{2} \) |
| 47 | \( 1 + (-6.50 - 6.15i)T + (2.54 + 46.9i)T^{2} \) |
| 53 | \( 1 + (-0.894 - 5.45i)T + (-50.2 + 16.9i)T^{2} \) |
| 61 | \( 1 + (4.24 - 3.22i)T + (16.3 - 58.7i)T^{2} \) |
| 67 | \( 1 + (4.27 + 10.7i)T + (-48.6 + 46.0i)T^{2} \) |
| 71 | \( 1 + (-5.14 - 4.87i)T + (3.84 + 70.8i)T^{2} \) |
| 73 | \( 1 + (1.61 + 0.355i)T + (66.2 + 30.6i)T^{2} \) |
| 79 | \( 1 + (-2.13 - 4.03i)T + (-44.3 + 65.3i)T^{2} \) |
| 83 | \( 1 + (-5.76 - 0.627i)T + (81.0 + 17.8i)T^{2} \) |
| 89 | \( 1 + (0.404 + 0.307i)T + (23.8 + 85.7i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 3.06i)T + (88.0 - 40.7i)T^{2} \) |
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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
−15.58061646377100510856438438073, −13.86564586798383251808605596505, −13.07054990335823766618303135441, −11.91313854536602354259733795716, −10.58733332151616882288782186541, −9.668059981838441384852792764452, −8.628330844631903851419111964734, −6.07751734579329155893090034461, −4.84907881048680702983407545800, −3.76239639381013006772060156308,
2.10730555801937065337033439949, 5.56609127240122851543321642825, 6.38174849997009468858734887407, 6.95399420568744451766215622451, 9.310047384854669316048257434038, 10.55880061241619031344113959729, 11.81108774344931069387261955711, 13.08071397515152982457805927364, 13.78542388049280715364294608957, 14.64456659301759829849810381814
