| L(s) = 1 | + (0.222 + 0.974i)2-s + (1.95 − 0.601i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (1.02 + 1.76i)6-s + (2.34 − 4.05i)7-s + (−0.623 − 0.781i)8-s + (0.965 − 0.658i)9-s + (0.988 + 0.149i)10-s + (1.10 + 0.530i)11-s + (−1.49 + 1.38i)12-s + (−3.31 + 0.499i)13-s + (4.47 + 1.38i)14-s + (0.152 − 2.03i)15-s + (0.623 − 0.781i)16-s + (−2.11 − 5.38i)17-s + ⋯ |
| L(s) = 1 | + (0.157 + 0.689i)2-s + (1.12 − 0.347i)3-s + (−0.450 + 0.216i)4-s + (0.163 − 0.416i)5-s + (0.416 + 0.721i)6-s + (0.885 − 1.53i)7-s + (−0.220 − 0.276i)8-s + (0.321 − 0.219i)9-s + (0.312 + 0.0471i)10-s + (0.331 + 0.159i)11-s + (−0.432 + 0.400i)12-s + (−0.919 + 0.138i)13-s + (1.19 + 0.369i)14-s + (0.0393 − 0.525i)15-s + (0.155 − 0.195i)16-s + (−0.513 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.15571 - 0.0667976i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.15571 - 0.0667976i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (-0.365 + 0.930i)T \) |
| 43 | \( 1 + (5.66 - 3.30i)T \) |
| good | 3 | \( 1 + (-1.95 + 0.601i)T + (2.47 - 1.68i)T^{2} \) |
| 7 | \( 1 + (-2.34 + 4.05i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.10 - 0.530i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (3.31 - 0.499i)T + (12.4 - 3.83i)T^{2} \) |
| 17 | \( 1 + (2.11 + 5.38i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-2.58 - 1.76i)T + (6.94 + 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.558 - 7.44i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (-8.01 - 2.47i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (3.78 - 3.51i)T + (2.31 - 30.9i)T^{2} \) |
| 37 | \( 1 + (0.0399 + 0.0692i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.47 - 10.8i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (-9.95 + 4.79i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (7.37 + 1.11i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (4.79 - 6.01i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-4.47 - 4.15i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (0.293 + 0.200i)T + (24.4 + 62.3i)T^{2} \) |
| 71 | \( 1 + (-0.716 + 9.56i)T + (-70.2 - 10.5i)T^{2} \) |
| 73 | \( 1 + (-1.28 + 0.193i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-2.04 + 3.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.34 + 1.33i)T + (68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (5.37 - 1.65i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + (10.4 + 5.02i)T + (60.4 + 75.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.20046445335014755910501611025, −9.896730313096285591498126862350, −9.195952273460471943709858165869, −8.157608523769678737146370560410, −7.46814821533339281244744848731, −6.95802574525003007663156860593, −5.18675660471813912592967859938, −4.41693394955332762957523150375, −3.13305195168061639780302701996, −1.41888766933557875419005394285,
2.18754053200515556955594135567, 2.69331003656872664223282976583, 4.05795383695230970578749423550, 5.18266421876743490082423739618, 6.31836336535899159205876287022, 7.940467315058771410064100532381, 8.691648644677056444367085565261, 9.219219125380937543473429273657, 10.24776140497455097034713412112, 11.16010360370185314906922662813
