| L(s) = 1 | + (0.114 − 0.169i)2-s + (−5.15 − 0.656i)3-s + (2.94 + 7.39i)4-s + (3.22 − 6.97i)5-s + (−0.702 + 0.797i)6-s + (−0.623 − 2.24i)7-s + (3.18 + 0.701i)8-s + (26.1 + 6.76i)9-s + (−0.810 − 1.34i)10-s + (−23.2 − 22.0i)11-s + (−10.3 − 40.0i)12-s + (−3.52 + 32.3i)13-s + (−0.451 − 0.152i)14-s + (−21.2 + 33.8i)15-s + (−45.7 + 43.3i)16-s + (72.8 + 20.2i)17-s + ⋯ |
| L(s) = 1 | + (0.0405 − 0.0598i)2-s + (−0.991 − 0.126i)3-s + (0.368 + 0.924i)4-s + (0.288 − 0.623i)5-s + (−0.0478 + 0.0542i)6-s + (−0.0336 − 0.121i)7-s + (0.140 + 0.0310i)8-s + (0.968 + 0.250i)9-s + (−0.0256 − 0.0426i)10-s + (−0.637 − 0.603i)11-s + (−0.248 − 0.963i)12-s + (−0.0751 + 0.691i)13-s + (−0.00862 − 0.00290i)14-s + (−0.365 + 0.582i)15-s + (−0.714 + 0.676i)16-s + (1.03 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.22915 + 0.590838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22915 + 0.590838i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.15 + 0.656i)T \) |
| 59 | \( 1 + (446. - 79.1i)T \) |
| good | 2 | \( 1 + (-0.114 + 0.169i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (-3.22 + 6.97i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (0.623 + 2.24i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (23.2 + 22.0i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (3.52 - 32.3i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (-72.8 - 20.2i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (-78.7 - 59.8i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-52.0 - 98.2i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (-0.985 + 0.668i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (-126. - 166. i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (0.00423 + 0.0192i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (-244. - 129. i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (73.0 + 77.0i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (304. - 140. i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (53.3 - 88.6i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-11.3 - 7.72i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (-102. + 463. i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (-89.6 - 193. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-97.0 + 288. i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-27.1 + 500. i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (-27.0 + 165. i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (-525. - 774. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (587. + 1.74e3i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−12.28304515915403892649198681168, −11.58699481336408823641798526679, −10.60836977309708170353790962645, −9.430826871558036214033400897903, −8.081606305674673829435085676246, −7.19837817242663158751889292879, −5.91968177790174607181305131585, −4.87383060933274765033309898084, −3.36574579043813033210832379969, −1.35701095304246114312113374657,
0.78263955319713881675927825532, 2.64952455058249501757838097954, 4.82108984901783038734080449557, 5.64800992967683756009984328075, 6.64165573811279008990863441397, 7.57789261557505685756824821959, 9.591052877781553997620828944267, 10.24254701290139686716747673131, 10.93808258538809298862823212612, 11.91958853907908822841739560866
