| L(s) = 1 | + (0.415 + 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (0.654 + 0.755i)6-s + (−0.257 − 1.79i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (2.50 − 5.48i)11-s + (−0.415 + 0.909i)12-s + (−0.456 + 3.17i)13-s + (1.52 − 0.977i)14-s + (−0.959 − 0.281i)15-s + (−0.142 − 0.989i)16-s + (2.98 + 3.44i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.376 − 0.241i)5-s + (0.267 + 0.308i)6-s + (−0.0973 − 0.676i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.0450 − 0.313i)10-s + (0.755 − 1.65i)11-s + (−0.119 + 0.262i)12-s + (−0.126 + 0.880i)13-s + (0.406 − 0.261i)14-s + (−0.247 − 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (0.724 + 0.835i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95908 - 0.111570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95908 - 0.111570i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-2.37 + 4.16i)T \) |
| good | 7 | \( 1 + (0.257 + 1.79i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.50 + 5.48i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.456 - 3.17i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.98 - 3.44i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.90 + 5.66i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.49 - 5.18i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (7.21 + 2.11i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.25 - 0.809i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-0.128 - 0.0823i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (4.19 - 1.23i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 4.07T + 47T^{2} \) |
| 53 | \( 1 + (0.323 + 2.24i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.64 - 11.4i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (14.6 + 4.30i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (1.63 + 3.58i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.08 - 2.36i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (1.57 - 1.81i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.792 + 5.51i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-1.26 + 0.815i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (4.20 - 1.23i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-11.9 - 7.67i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.47501991954241116499469589172, −9.046998233656855429631992603310, −8.852089196599045925869147112957, −7.73824157592196990467513769385, −6.98712027910848172889528715706, −6.15878823078141159883719452936, −4.93481275986860233830737339465, −3.85938280522525683223407278174, −3.13203361952038390426545233857, −1.02474421154108515891719225297,
1.60733557422886542667818038808, 2.89192986393862966859748232010, 3.70569068920039563286606923038, 4.86009908732182959209109693148, 5.75160611516279505971681640694, 7.20938392762715379255491428094, 7.80576154625141225313966406339, 9.068008812125389676071734436566, 9.725789897970629102352589800841, 10.27629436211919966400467972527
