| L(s) = 1 | + (0.679 + 1.33i)2-s + (−2.32 − 1.18i)3-s + (−0.139 + 0.192i)4-s + (−0.203 − 2.22i)5-s − 3.90i·6-s + (1.63 − 0.259i)7-s + (2.60 + 0.412i)8-s + (2.25 + 3.09i)9-s + (2.82 − 1.78i)10-s + (2.32 − 3.20i)11-s + (0.552 − 0.281i)12-s + (−4.20 − 2.14i)13-s + (1.45 + 2.00i)14-s + (−2.16 + 5.42i)15-s + (1.36 + 4.20i)16-s + (−2.82 − 0.447i)17-s + ⋯ |
| L(s) = 1 | + (0.480 + 0.942i)2-s + (−1.34 − 0.685i)3-s + (−0.0697 + 0.0960i)4-s + (−0.0910 − 0.995i)5-s − 1.59i·6-s + (0.618 − 0.0979i)7-s + (0.920 + 0.145i)8-s + (0.750 + 1.03i)9-s + (0.894 − 0.564i)10-s + (0.702 − 0.966i)11-s + (0.159 − 0.0813i)12-s + (−1.16 − 0.594i)13-s + (0.389 + 0.535i)14-s + (−0.559 + 1.40i)15-s + (0.341 + 1.05i)16-s + (−0.684 − 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02791 - 0.245562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02791 - 0.245562i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.203 + 2.22i)T \) |
| 31 | \( 1 + (-4.64 - 3.06i)T \) |
| good | 2 | \( 1 + (-0.679 - 1.33i)T + (-1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (2.32 + 1.18i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-1.63 + 0.259i)T + (6.65 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-2.32 + 3.20i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (4.20 + 2.14i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (2.82 + 0.447i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-2.89 - 0.941i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.889 - 5.61i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.33 + 4.11i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-5.18 - 5.18i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.463 - 1.42i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.95 - 5.80i)T + (-25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (0.940 + 0.479i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-2.20 + 13.9i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.67 + 0.870i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 - 14.4iT - 61T^{2} \) |
| 67 | \( 1 + (5.73 - 5.73i)T - 67iT^{2} \) |
| 71 | \( 1 + (-0.337 + 0.244i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.33 + 0.211i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (1.58 - 1.15i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.865 - 1.69i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (8.53 + 6.20i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-13.2 + 2.09i)T + (92.2 - 29.9i)T^{2} \) |
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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
−13.00264296566205318792370579485, −11.76675204334111654646835481161, −11.39829696511838476703349373333, −9.924833650111526739897836678841, −8.251817445870048565097193166894, −7.34620743001771114117710322852, −6.19853412528161231247858947720, −5.39927478132799013574094134638, −4.59534720065959757695610930638, −1.19180734618107664451803022946,
2.33180679180089323183135397076, 4.17764179931994912581315307526, 4.85333095171432082398273390584, 6.50406011507139197379124179992, 7.44305677984145244235805924527, 9.573129771251803961913310170556, 10.46732901058145340622189003643, 11.14758635358571119734256438438, 11.88963000336077776788307983417, 12.39914924887804011023096778701
