| L(s) = 1 | + (1.27 − 2.21i)2-s + (0.130 − 0.401i)3-s + (−2.25 − 3.91i)4-s + (−0.658 + 0.293i)5-s + (−0.720 − 0.800i)6-s + (−1.31 + 0.585i)7-s − 6.42·8-s + (2.28 + 1.65i)9-s + (−0.192 + 1.83i)10-s + (2.41 + 0.512i)11-s + (−1.86 + 0.396i)12-s + (3.18 − 0.677i)13-s + (−0.384 + 3.65i)14-s + (0.0317 + 0.302i)15-s + (−3.68 + 6.37i)16-s + (−0.430 − 4.09i)17-s + ⋯ |
| L(s) = 1 | + (0.902 − 1.56i)2-s + (0.0752 − 0.231i)3-s + (−1.12 − 1.95i)4-s + (−0.294 + 0.131i)5-s + (−0.294 − 0.326i)6-s + (−0.497 + 0.221i)7-s − 2.27·8-s + (0.761 + 0.552i)9-s + (−0.0608 + 0.579i)10-s + (0.727 + 0.154i)11-s + (−0.537 + 0.114i)12-s + (0.884 − 0.187i)13-s + (−0.102 + 0.977i)14-s + (0.00821 + 0.0781i)15-s + (−0.920 + 1.59i)16-s + (−0.104 − 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.643645 - 1.46094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643645 - 1.46094i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (12.1 + 1.63i)T \) |
| good | 2 | \( 1 + (-1.27 + 2.21i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.130 + 0.401i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (0.658 - 0.293i)T + (3.34 - 3.71i)T^{2} \) |
| 7 | \( 1 + (1.31 - 0.585i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-2.41 - 0.512i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.18 + 0.677i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (0.430 + 4.09i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 + (3.07 - 5.32i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.584 - 1.80i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.762 - 7.25i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (0.484 + 0.538i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-2.13 + 6.56i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (8.39 + 3.73i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-8.77 + 1.86i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-3.10 - 9.55i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + (-8.09 + 8.98i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (10.0 - 7.32i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (0.377 - 3.59i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (8.63 + 6.27i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (6.02 + 4.37i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.67 + 1.94i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.66 - 1.63i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (0.955 + 9.08i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.51926463929012596632484314864, −11.80277035433953043451432238324, −10.85812400069279611727383572309, −9.955745012159409509287402072123, −8.923411831953293633541725466274, −7.15243650635674355062810425154, −5.67142053441182899231848707416, −4.30934746542587489438667809937, −3.26168714358050039527391090861, −1.66042723473259324287614506573,
3.83119830253433482195904492006, 4.30479285096848399228916705122, 6.17723108881166090436192812484, 6.55190266294655827197424464329, 7.964883854959881535479390211862, 8.818411379320940729018579736399, 10.13392536387041806966904787496, 11.80451313244303005856563887839, 12.83005992593490984619265847385, 13.46054995523038793028616144373
