| L(s) = 1 | + (1.57 + 0.759i)2-s + (0.234 + 1.87i)3-s + (0.661 + 0.830i)4-s + (−1.51 − 0.228i)5-s + (−1.05 + 3.13i)6-s + (0.866 − 0.175i)7-s + (−0.365 − 1.60i)8-s + (−0.545 + 0.138i)9-s + (−2.21 − 1.51i)10-s + (−0.904 − 1.25i)11-s + (−1.39 + 1.43i)12-s + (1.48 + 1.67i)13-s + (1.49 + 0.381i)14-s + (0.0721 − 2.89i)15-s + (1.11 − 4.87i)16-s + (−0.105 − 4.21i)17-s + ⋯ |
| L(s) = 1 | + (1.11 + 0.536i)2-s + (0.135 + 1.08i)3-s + (0.330 + 0.415i)4-s + (−0.677 − 0.102i)5-s + (−0.429 + 1.27i)6-s + (0.327 − 0.0662i)7-s + (−0.129 − 0.565i)8-s + (−0.181 + 0.0463i)9-s + (−0.700 − 0.477i)10-s + (−0.272 − 0.379i)11-s + (−0.403 + 0.414i)12-s + (0.410 + 0.465i)13-s + (0.400 + 0.102i)14-s + (0.0186 − 0.746i)15-s + (0.277 − 1.21i)16-s + (−0.0254 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38028 + 0.974697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38028 + 0.974697i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 127 | \( 1 + (8.52 + 7.37i)T \) |
| good | 2 | \( 1 + (-1.57 - 0.759i)T + (1.24 + 1.56i)T^{2} \) |
| 3 | \( 1 + (-0.234 - 1.87i)T + (-2.90 + 0.740i)T^{2} \) |
| 5 | \( 1 + (1.51 + 0.228i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.175i)T + (6.45 - 2.71i)T^{2} \) |
| 11 | \( 1 + (0.904 + 1.25i)T + (-3.50 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.48 - 1.67i)T + (-1.61 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.105 + 4.21i)T + (-16.9 + 0.847i)T^{2} \) |
| 19 | \( 1 + (3.00 - 5.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.01 + 1.40i)T + (-7.32 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.860 + 0.0429i)T + (28.8 + 2.88i)T^{2} \) |
| 31 | \( 1 + (-2.04 - 1.24i)T + (14.1 + 27.5i)T^{2} \) |
| 37 | \( 1 + (8.04 + 2.92i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-0.988 + 0.603i)T + (18.7 - 36.4i)T^{2} \) |
| 43 | \( 1 + (4.57 - 7.08i)T + (-17.6 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-1.72 + 4.38i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-2.99 - 3.07i)T + (-1.32 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-0.725 + 0.609i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.493 - 0.152i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 7.95i)T + (18.1 - 64.4i)T^{2} \) |
| 71 | \( 1 + (-4.29 - 9.51i)T + (-46.9 + 53.2i)T^{2} \) |
| 73 | \( 1 + (0.780 - 10.4i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-14.1 - 1.41i)T + (77.4 + 15.6i)T^{2} \) |
| 83 | \( 1 + (0.942 - 3.34i)T + (-70.8 - 43.2i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 8.59i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (4.59 - 8.96i)T + (-56.6 - 78.7i)T^{2} \) |
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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
−13.89203294509604253356423621205, −12.69942441111918944006128377644, −11.66366211858880228523030478969, −10.50631760709105449144829130373, −9.427010534514865329268365213789, −8.146251707065150095007897132369, −6.75462327638864303002776712728, −5.34887492144032195792355234869, −4.35970949719473416064297995927, −3.54030763160515141668800001556,
2.08352639228901209185716452682, 3.66042766360570418906427462726, 4.96841250699696497350026085820, 6.45621538039117357957705976985, 7.72365291233478105564528649960, 8.560216664543813808578031602725, 10.55918029670495301021893572947, 11.53787142696349952197604186458, 12.37015928477294510800465451558, 13.11254528016262941395789704281
