| L(s) = 1 | + (0.0713 − 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (−2.07 + 0.840i)5-s + (−0.142 − 0.989i)6-s + (1.67 − 0.913i)7-s + (−0.212 + 0.977i)8-s + (0.909 − 0.415i)9-s + (0.690 + 2.12i)10-s + (−3.05 − 2.64i)11-s + (−0.997 + 0.0713i)12-s + (0.560 − 1.02i)13-s + (−0.792 − 1.73i)14-s + (−1.84 + 1.26i)15-s + (0.959 + 0.281i)16-s + (−5.42 − 4.06i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.926 + 0.376i)5-s + (−0.0580 − 0.404i)6-s + (0.632 − 0.345i)7-s + (−0.0751 + 0.345i)8-s + (0.303 − 0.138i)9-s + (0.218 + 0.672i)10-s + (−0.921 − 0.798i)11-s + (−0.287 + 0.0205i)12-s + (0.155 − 0.284i)13-s + (−0.211 − 0.463i)14-s + (−0.476 + 0.325i)15-s + (0.239 + 0.0704i)16-s + (−1.31 − 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.384620 - 1.13099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.384620 - 1.13099i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (2.07 - 0.840i)T \) |
| 23 | \( 1 + (-4.56 - 1.45i)T \) |
| good | 7 | \( 1 + (-1.67 + 0.913i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (3.05 + 2.64i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.560 + 1.02i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (5.42 + 4.06i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.975 + 6.78i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.845 - 0.121i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.124 + 0.0798i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (7.03 + 2.62i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-2.18 + 4.77i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.188 + 0.868i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (3.33 + 3.33i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.25 - 4.12i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-3.50 - 11.9i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.84 - 2.86i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-5.79 - 0.414i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (5.30 + 6.12i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.688 - 0.919i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-8.18 + 2.40i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (3.56 - 9.55i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.85 + 1.19i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.84 + 4.95i)T + (-73.3 + 63.5i)T^{2} \) |
| 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
−10.43915761813973912886255025032, −9.059361088381692361757337754064, −8.591122137502729990557721366605, −7.55006419838767149896977501583, −6.94554265555887062961037045677, −5.23211035998965870839940949151, −4.40880597051171443948473287174, −3.24683223128819010342898843140, −2.48591138282285971395870421178, −0.58113161769783568817251554086,
1.88091521398251257723059691387, 3.51530276491800145197808389723, 4.51170172514203026085359233965, 5.18139944074155940128148972629, 6.55227703506169443730109092263, 7.52709197575231648684145269178, 8.266129316230615903970608997645, 8.656019030772838992577867237241, 9.781898541915535762524476753910, 10.74896732431327223889941705009
