| L(s) = 1 | + (−0.150 + 1.40i)2-s + (0.269 − 0.830i)3-s + (−1.95 − 0.423i)4-s + (−2.21 − 0.294i)5-s + (1.12 + 0.504i)6-s + (0.0768 − 0.0768i)7-s + (0.890 − 2.68i)8-s + (1.81 + 1.31i)9-s + (0.748 − 3.07i)10-s + (5.02 + 0.795i)11-s + (−0.878 + 1.50i)12-s + (1.22 − 1.68i)13-s + (0.0964 + 0.119i)14-s + (−0.842 + 1.76i)15-s + (3.64 + 1.65i)16-s + (1.70 + 0.869i)17-s + ⋯ |
| L(s) = 1 | + (−0.106 + 0.994i)2-s + (0.155 − 0.479i)3-s + (−0.977 − 0.211i)4-s + (−0.991 − 0.131i)5-s + (0.459 + 0.205i)6-s + (0.0290 − 0.0290i)7-s + (0.314 − 0.949i)8-s + (0.603 + 0.438i)9-s + (0.236 − 0.971i)10-s + (1.51 + 0.239i)11-s + (−0.253 + 0.435i)12-s + (0.339 − 0.467i)13-s + (0.0257 + 0.0319i)14-s + (−0.217 + 0.454i)15-s + (0.910 + 0.414i)16-s + (0.413 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11535 + 0.451598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11535 + 0.451598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.150 - 1.40i)T \) |
| 5 | \( 1 + (2.21 + 0.294i)T \) |
| good | 3 | \( 1 + (-0.269 + 0.830i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.0768 + 0.0768i)T - 7iT^{2} \) |
| 11 | \( 1 + (-5.02 - 0.795i)T + (10.4 + 3.39i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.68i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.70 - 0.869i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.02 - 2.00i)T + (-11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-4.91 - 0.779i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.862 + 1.69i)T + (-17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (5.10 - 1.66i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.25 + 5.84i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.41 + 3.31i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 0.756iT - 43T^{2} \) |
| 47 | \( 1 + (0.133 - 0.0682i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-3.02 + 9.29i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (2.67 - 0.424i)T + (56.1 - 18.2i)T^{2} \) |
| 61 | \( 1 + (-3.20 - 0.508i)T + (58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (14.2 - 4.64i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.89 - 8.89i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.822 - 5.19i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (2.66 - 8.19i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.83 + 5.64i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.76 + 5.63i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (6.81 + 13.3i)T + (-57.0 + 78.4i)T^{2} \) |
| 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
−11.45792378608380942361821373755, −10.36121059932167835320783674394, −9.204968846021065039986962042765, −8.444118877585296088993549919209, −7.48354622587580094423416416246, −7.00503339260643623231187268833, −5.82093179500316242573453549604, −4.49098165539760111920064281984, −3.67693565023307897540390833134, −1.20613126493677410901123747606,
1.18215184691212411708035459932, 3.18577603749872260793395869990, 3.97213306871568455633578552626, 4.73513813522379921655473357317, 6.51230603916491373557699383406, 7.60545045028617743287417950655, 8.969472709465928542745129119929, 9.165269463970632415433781121793, 10.39599078426269554711564852552, 11.26548726498559119530990195890
