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
−13.41699243860419838552019021539, −12.90874311609335777891448447617, −11.92244597716342762386893056639, −10.45644728872431268883259492838, −9.040877069803952193906867212561, −8.155837314410357668406601721706, −6.66393955083290332549910886320, −5.36043344164897587173746029547, −4.44375388383588924776995605561, −2.39924828180303328060907750517,
2.94203416129092519818582928257, 4.03553998627969204557224569917, 5.21203818388439002762225678670, 6.81213219605916437130063639835, 8.238337850565336716180557979955, 9.420192831193912733743114479277, 10.64112511955173776146708047952, 11.58846661311738863973234501765, 12.80951518135256025795720950942, 13.62944699557183762102929708012