| L(s) = 1 | + (−1.16 + 0.794i)2-s + (−0.623 − 0.244i)3-s + (0.737 − 1.85i)4-s + (2.60 − 0.195i)5-s + (0.924 − 0.209i)6-s + (−0.0646 + 0.111i)7-s + (0.613 + 2.76i)8-s + (−1.86 − 1.73i)9-s + (−2.88 + 2.29i)10-s + (3.12 + 0.713i)11-s + (−0.915 + 0.979i)12-s + (−1.00 − 1.47i)13-s + (−0.0133 − 0.182i)14-s + (−1.67 − 0.515i)15-s + (−2.91 − 2.74i)16-s + (0.593 − 7.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.827 + 0.561i)2-s + (−0.360 − 0.141i)3-s + (0.368 − 0.929i)4-s + (1.16 − 0.0872i)5-s + (0.377 − 0.0853i)6-s + (−0.0244 + 0.0423i)7-s + (0.217 + 0.976i)8-s + (−0.623 − 0.578i)9-s + (−0.913 + 0.725i)10-s + (0.943 + 0.215i)11-s + (−0.264 + 0.282i)12-s + (−0.279 − 0.410i)13-s + (−0.00355 − 0.0487i)14-s + (−0.431 − 0.133i)15-s + (−0.727 − 0.685i)16-s + (0.143 − 1.92i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.958957 - 0.105738i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.958957 - 0.105738i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 - 0.794i)T \) |
| 43 | \( 1 + (5.87 - 2.91i)T \) |
| good | 3 | \( 1 + (0.623 + 0.244i)T + (2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (-2.60 + 0.195i)T + (4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (0.0646 - 0.111i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.12 - 0.713i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (1.00 + 1.47i)T + (-4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.593 + 7.91i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-2.92 - 3.15i)T + (-1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.75 - 0.541i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-4.95 + 1.94i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-7.84 - 1.18i)T + (29.6 + 9.13i)T^{2} \) |
| 37 | \( 1 + (-9.30 + 5.36i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.85 - 3.58i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (1.81 + 7.93i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.800 + 1.17i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (1.38 + 2.87i)T + (-36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-0.533 - 3.54i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (2.97 + 3.21i)T + (-5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (11.1 + 3.42i)T + (58.6 + 39.9i)T^{2} \) |
| 73 | \( 1 + (10.9 - 7.46i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-3.16 + 5.47i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.96 - 1.16i)T + (60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (2.41 - 6.14i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.24 + 5.47i)T + (-87.3 - 42.0i)T^{2} \) |
| 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
−11.58593752164356766242421398565, −10.10676274332417377164796678502, −9.645357506754742762877723649796, −8.880730468477175619302601923518, −7.65315523695904372973231049563, −6.54880943881788954060658559462, −5.92085419165297587230512803144, −4.96006298487593939430496056895, −2.70970558762737545921105904927, −1.05221466845920731690035862888,
1.53679140657492645774077765876, 2.81863215621065553690261840967, 4.39335191113837410063083073615, 5.94009460296805616426718522352, 6.63809205681165267539205857281, 8.108677189498839486684841642283, 8.886847584193866833227468166749, 9.913305011443892730970590362583, 10.41908867081506229412868872937, 11.44729789915664517959798154637
