| L(s) = 1 | + (0.377 − 1.65i)2-s + (0.955 + 0.294i)3-s + (−0.793 − 0.382i)4-s + (1.06 + 2.70i)5-s + (0.848 − 1.46i)6-s + (−1.18 − 2.05i)7-s + (1.18 − 1.48i)8-s + (0.826 + 0.563i)9-s + (4.87 − 0.734i)10-s + (−2.66 + 1.28i)11-s + (−0.645 − 0.598i)12-s + (−7.10 − 1.07i)13-s + (−3.85 + 1.18i)14-s + (0.216 + 2.89i)15-s + (−3.10 − 3.89i)16-s + (−1.89 + 4.83i)17-s + ⋯ |
| L(s) = 1 | + (0.267 − 1.17i)2-s + (0.551 + 0.170i)3-s + (−0.396 − 0.191i)4-s + (0.474 + 1.20i)5-s + (0.346 − 0.600i)6-s + (−0.449 − 0.778i)7-s + (0.418 − 0.525i)8-s + (0.275 + 0.187i)9-s + (1.54 − 0.232i)10-s + (−0.804 + 0.387i)11-s + (−0.186 − 0.172i)12-s + (−1.97 − 0.297i)13-s + (−1.03 + 0.317i)14-s + (0.0560 + 0.747i)15-s + (−0.777 − 0.974i)16-s + (−0.459 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28945 - 0.684256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28945 - 0.684256i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.955 - 0.294i)T \) |
| 43 | \( 1 + (2.81 - 5.92i)T \) |
| good | 2 | \( 1 + (-0.377 + 1.65i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (-1.06 - 2.70i)T + (-3.66 + 3.40i)T^{2} \) |
| 7 | \( 1 + (1.18 + 2.05i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.66 - 1.28i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (7.10 + 1.07i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (1.89 - 4.83i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-6.35 + 4.33i)T + (6.94 - 17.6i)T^{2} \) |
| 23 | \( 1 + (0.0233 - 0.311i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-3.38 + 1.04i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (-1.17 - 1.09i)T + (2.31 + 30.9i)T^{2} \) |
| 37 | \( 1 + (2.20 - 3.81i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.651 + 2.85i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (-4.97 - 2.39i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (8.76 - 1.32i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-5.57 - 6.99i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-4.88 + 4.53i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (2.08 - 1.41i)T + (24.4 - 62.3i)T^{2} \) |
| 71 | \( 1 + (0.748 + 9.98i)T + (-70.2 + 10.5i)T^{2} \) |
| 73 | \( 1 + (-6.97 - 1.05i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (4.10 + 7.11i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.93 - 2.44i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (5.62 + 1.73i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + (-0.681 + 0.328i)T + (60.4 - 75.8i)T^{2} \) |
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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
−13.17110254849673306108962888712, −12.18174370079501349762374013936, −10.87792620229821601944554149745, −10.15925099617663285249984926004, −9.701183383780595737191813397299, −7.57176170650006130332247051020, −6.86059832507052857214790058877, −4.74419209738567529216937875737, −3.19094633841378155102819598638, −2.41871965261502672234187486440,
2.46061417948640382216714990334, 4.95080893913013935991465668066, 5.51337062435665206903229908800, 7.04233405886997364857546421875, 7.997027642964701117716825094886, 9.084576075176022749225820468334, 9.883927817804439356742885884544, 11.87166969250693236983444650201, 12.71305658124637742467407274534, 13.75242558734214644634099392492
