| L(s) = 1 | + (0.186 − 0.215i)2-s + (1.19 − 0.770i)3-s + (0.273 + 1.89i)4-s + (−0.415 − 0.909i)5-s + (0.0577 − 0.401i)6-s + (1.34 − 0.393i)7-s + (0.938 + 0.603i)8-s + (−0.403 + 0.883i)9-s + (−0.273 − 0.0801i)10-s + (−3.15 − 3.64i)11-s + (1.79 + 2.06i)12-s + (0.273 + 0.0804i)13-s + (0.165 − 0.361i)14-s + (−1.19 − 0.770i)15-s + (−3.37 + 0.991i)16-s + (0.0679 − 0.472i)17-s + ⋯ |
| L(s) = 1 | + (0.131 − 0.152i)2-s + (0.691 − 0.444i)3-s + (0.136 + 0.949i)4-s + (−0.185 − 0.406i)5-s + (0.0235 − 0.163i)6-s + (0.506 − 0.148i)7-s + (0.331 + 0.213i)8-s + (−0.134 + 0.294i)9-s + (−0.0863 − 0.0253i)10-s + (−0.952 − 1.09i)11-s + (0.516 + 0.596i)12-s + (0.0759 + 0.0223i)13-s + (0.0441 − 0.0966i)14-s + (−0.309 − 0.198i)15-s + (−0.844 + 0.247i)16-s + (0.0164 − 0.114i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31761 - 0.121376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31761 - 0.121376i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.63 - 1.21i)T \) |
| good | 2 | \( 1 + (-0.186 + 0.215i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (-1.19 + 0.770i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 0.393i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (3.15 + 3.64i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.273 - 0.0804i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.0679 + 0.472i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.140 - 0.979i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.0219 + 0.152i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (3.01 + 1.93i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.70 - 3.74i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.86 + 6.27i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-9.05 + 5.81i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 9.42T + 47T^{2} \) |
| 53 | \( 1 + (-4.62 + 1.35i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-6.57 - 1.92i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-9.16 - 5.89i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.62 + 5.33i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-7.91 + 9.13i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.970 - 6.75i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (11.4 + 3.37i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.70 + 5.91i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (9.20 - 5.91i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (6.50 + 14.2i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−13.55632801952472712886422206796, −12.66596727262555953019655040313, −11.59924048119489735856573047498, −10.64108591670488987690767781852, −8.788568191691154753155280632593, −8.120345013573138691426651627634, −7.39200149738244285679628113725, −5.42817525543431046246311918879, −3.79008943194424780498855567302, −2.38149058812684588018471157631,
2.37342495972559384238638693814, 4.24588047648859084775665155730, 5.55019188116241483033727476684, 6.97323951887452253014065816000, 8.217230831644543047946606382735, 9.541340541845704309036027880977, 10.29490646556478082335231114288, 11.34117344168438139771595755366, 12.67014481011005280047626521282, 14.05467231077913041445606932227
