| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (0.172 − 0.975i)5-s + (0.173 + 0.984i)6-s + (−1.52 − 2.16i)7-s + (0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.758 + 0.636i)10-s + (2.52 + 4.36i)11-s + (0.500 − 0.866i)12-s + (1.12 + 6.37i)13-s + (−0.222 + 2.63i)14-s + (−0.758 + 0.636i)15-s + (−0.939 + 0.342i)16-s + (−0.721 + 4.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.442 − 0.371i)3-s + (0.0868 + 0.492i)4-s + (0.0769 − 0.436i)5-s + (0.0708 + 0.402i)6-s + (−0.576 − 0.817i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.239 + 0.201i)10-s + (0.760 + 1.31i)11-s + (0.144 − 0.249i)12-s + (0.311 + 1.76i)13-s + (−0.0594 + 0.704i)14-s + (−0.195 + 0.164i)15-s + (−0.234 + 0.0855i)16-s + (−0.175 + 0.992i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.856836 + 0.0448113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.856836 + 0.0448113i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (1.52 + 2.16i)T \) |
| 19 | \( 1 + (2.32 + 3.68i)T \) |
| good | 5 | \( 1 + (-0.172 + 0.975i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-2.52 - 4.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.12 - 6.37i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.721 - 4.09i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (5.01 + 1.82i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (5.06 + 1.84i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + (-4.25 - 7.37i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.68 + 9.54i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.66 - 4.75i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.195 - 1.10i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.39 - 7.90i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.363 - 2.06i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.64 - 1.69i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.01 - 0.853i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.55 - 4.65i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (5.98 + 5.02i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 4.11i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.13 - 5.43i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.67 + 1.40i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.0777 + 0.0283i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−10.24576186305012725711998718340, −9.473965320039319741309905022948, −8.815160051253617237841508865198, −7.67903328504139796389053602627, −6.72214664127313669982176531703, −6.35740428693961776421050897827, −4.42805744766560754588444060143, −4.15056047791423078843233434119, −2.24499418912479039579655758572, −1.19320746614465257008732405303,
0.63471231770503243964469601608, 2.70903572117235043450009806596, 3.70442563082668999367849552536, 5.30444327620796345127556076771, 6.01288371853762563450803464075, 6.49022691848702892250087418217, 7.85499219147117231435595960094, 8.553607208995179661760162136602, 9.445526644249827493109849853315, 10.16781156941289209879115502165
