| L(s) = 1 | + (0.212 − 0.977i)2-s + (2.68 − 2.01i)3-s + (−0.909 − 0.415i)4-s + (−1.91 − 1.15i)5-s + (−1.39 − 3.05i)6-s + (−0.286 + 4.01i)7-s + (−0.599 + 0.800i)8-s + (2.32 − 7.92i)9-s + (−1.53 + 1.62i)10-s + (0.608 − 0.946i)11-s + (−3.27 + 0.713i)12-s + (1.75 − 0.125i)13-s + (3.85 + 1.13i)14-s + (−7.46 + 0.742i)15-s + (0.654 + 0.755i)16-s + (0.281 − 0.755i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 0.690i)2-s + (1.55 − 1.16i)3-s + (−0.454 − 0.207i)4-s + (−0.855 − 0.517i)5-s + (−0.569 − 1.24i)6-s + (−0.108 + 1.51i)7-s + (−0.211 + 0.283i)8-s + (0.776 − 2.64i)9-s + (−0.485 + 0.513i)10-s + (0.183 − 0.285i)11-s + (−0.946 + 0.205i)12-s + (0.485 − 0.0347i)13-s + (1.03 + 0.302i)14-s + (−1.92 + 0.191i)15-s + (0.163 + 0.188i)16-s + (0.0683 − 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01223 - 1.44762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01223 - 1.44762i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.212 + 0.977i)T \) |
| 5 | \( 1 + (1.91 + 1.15i)T \) |
| 23 | \( 1 + (-4.13 - 2.43i)T \) |
| good | 3 | \( 1 + (-2.68 + 2.01i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (0.286 - 4.01i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (-0.608 + 0.946i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.75 + 0.125i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-0.281 + 0.755i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (2.21 - 4.85i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.62 - 1.19i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.498 - 3.46i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.15 + 2.11i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-2.27 + 0.667i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (2.49 + 3.32i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-2.95 + 2.95i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.73 + 0.481i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-2.62 - 2.27i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-11.6 + 1.67i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.29 + 0.282i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (12.9 - 8.34i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (6.76 - 2.52i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (6.05 - 6.98i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.94 + 2.15i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (0.442 - 3.07i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (16.3 + 8.91i)T + (52.4 + 81.6i)T^{2} \) |
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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
−12.29566436349161701761491193429, −11.38927181441270716460913943587, −9.579801760405756454305383478002, −8.613351736266816198894460413138, −8.470034309953444829171551971786, −7.14753570830982016247547110529, −5.70528538229979181317540171602, −3.81997217323124138963180711079, −2.88334894813737550500240196224, −1.55721916285357693446568403082,
3.04428243140719078536068957490, 4.05346794474412131173543740485, 4.57923768870602046526170720553, 6.86068124355821409822035403734, 7.63286893403229369675308969133, 8.455663466616126959193732719380, 9.442565049044605773068777104739, 10.45274119390691100358641543458, 11.10683448764081577142964123556, 13.09150515872792022044613238193
