| L(s) = 1 | + (0.588 − 0.679i)2-s + (−1.58 + 1.01i)3-s + (0.169 + 1.17i)4-s + (0.415 + 0.909i)5-s + (−0.240 + 1.67i)6-s + (2.75 − 0.807i)7-s + (2.41 + 1.55i)8-s + (0.220 − 0.482i)9-s + (0.862 + 0.253i)10-s + (0.529 + 0.610i)11-s + (−1.46 − 1.69i)12-s + (−5.30 − 1.55i)13-s + (1.07 − 2.34i)14-s + (−1.58 − 1.01i)15-s + (0.187 − 0.0550i)16-s + (0.945 − 6.57i)17-s + ⋯ |
| L(s) = 1 | + (0.416 − 0.480i)2-s + (−0.912 + 0.586i)3-s + (0.0848 + 0.589i)4-s + (0.185 + 0.406i)5-s + (−0.0981 + 0.682i)6-s + (1.03 − 0.305i)7-s + (0.853 + 0.548i)8-s + (0.0734 − 0.160i)9-s + (0.272 + 0.0800i)10-s + (0.159 + 0.184i)11-s + (−0.423 − 0.488i)12-s + (−1.47 − 0.431i)13-s + (0.286 − 0.626i)14-s + (−0.408 − 0.262i)15-s + (0.0468 − 0.0137i)16-s + (0.229 − 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04873 + 0.320575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04873 + 0.320575i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-2.45 + 4.12i)T \) |
| good | 2 | \( 1 + (-0.588 + 0.679i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (1.58 - 1.01i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-2.75 + 0.807i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.529 - 0.610i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (5.30 + 1.55i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.945 + 6.57i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.395 - 2.74i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.755 + 5.25i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.97 - 3.83i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (2.16 - 4.73i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.70 + 5.91i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.67 + 1.07i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 + (-3.88 + 1.14i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.00 - 0.589i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (10.3 + 6.64i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (6.44 - 7.44i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (1.55 - 1.79i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.06 - 7.40i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-9.71 - 2.85i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.53 - 3.35i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (6.38 - 4.10i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.26 + 4.96i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−13.72194095589619237915757694071, −12.17886564952292561186223138054, −11.72177059472140611636023004471, −10.73971564792227001317264469586, −9.930223988190721947256659159535, −8.094178753835703749463906587053, −7.07603145430125648400698413605, −5.17856108926191230092821772416, −4.52856700341064714048980597946, −2.64245597640926023958474359351,
1.54328572897913314404064380322, 4.69143374582617444253140930982, 5.51719609948845399620509211323, 6.53495287070391705606803456911, 7.67204683685654727417528243156, 9.205916354420590320577123678907, 10.55601834891827767001052896488, 11.57205236806640663056482265795, 12.41349711440658320166238350807, 13.49067716168328750136236446557
