| L(s) = 1 | + (1.31 − 0.386i)2-s + (0.654 + 0.755i)3-s + (−0.0971 + 0.0624i)4-s + (0.142 + 0.989i)5-s + (1.15 + 0.742i)6-s + (0.925 − 2.02i)7-s + (−1.90 + 2.19i)8-s + (0.284 − 1.97i)9-s + (0.570 + 1.24i)10-s + (−1.98 − 0.584i)11-s + (−0.110 − 0.0325i)12-s + (−0.425 − 0.932i)13-s + (0.435 − 3.02i)14-s + (−0.654 + 0.755i)15-s + (−1.56 + 3.41i)16-s + (−3.33 − 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.931 − 0.273i)2-s + (0.378 + 0.436i)3-s + (−0.0485 + 0.0312i)4-s + (0.0636 + 0.442i)5-s + (0.471 + 0.302i)6-s + (0.349 − 0.766i)7-s + (−0.672 + 0.775i)8-s + (0.0948 − 0.659i)9-s + (0.180 + 0.394i)10-s + (−0.599 − 0.176i)11-s + (−0.0319 − 0.00939i)12-s + (−0.118 − 0.258i)13-s + (0.116 − 0.809i)14-s + (−0.169 + 0.195i)15-s + (−0.390 + 0.854i)16-s + (−0.807 − 0.519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59956 + 0.100413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59956 + 0.100413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-2.01 - 4.35i)T \) |
| good | 2 | \( 1 + (-1.31 + 0.386i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (-0.654 - 0.755i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.925 + 2.02i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (1.98 + 0.584i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.425 + 0.932i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (3.33 + 2.14i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.879 - 0.565i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.95 - 3.18i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.78 - 3.21i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.851 + 5.92i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.36 - 9.47i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (1.23 + 1.42i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + (-4.95 + 10.8i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.75 - 10.4i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-4.20 + 4.85i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.93 + 1.15i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (9.69 - 2.84i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-7.65 + 4.91i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.59 - 3.48i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.03 - 7.20i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.80 - 7.85i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.207 + 1.44i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.62944699557183762102929708012, −12.80951518135256025795720950942, −11.58846661311738863973234501765, −10.64112511955173776146708047952, −9.420192831193912733743114479277, −8.238337850565336716180557979955, −6.81213219605916437130063639835, −5.21203818388439002762225678670, −4.03553998627969204557224569917, −2.94203416129092519818582928257,
2.39924828180303328060907750517, 4.44375388383588924776995605561, 5.36043344164897587173746029547, 6.66393955083290332549910886320, 8.155837314410357668406601721706, 9.040877069803952193906867212561, 10.45644728872431268883259492838, 11.92244597716342762386893056639, 12.90874311609335777891448447617, 13.41699243860419838552019021539
