| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (1.41 + 0.910i)5-s + (−0.654 − 0.755i)6-s + (−0.142 − 0.989i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.239 + 1.66i)10-s + (1.93 − 4.24i)11-s + (0.415 − 0.909i)12-s + (0.837 − 5.82i)13-s + (0.841 − 0.540i)14-s + (−1.61 − 0.474i)15-s + (−0.142 − 0.989i)16-s + (−4.05 − 4.68i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.633 + 0.407i)5-s + (−0.267 − 0.308i)6-s + (−0.0537 − 0.374i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0758 + 0.527i)10-s + (0.584 − 1.27i)11-s + (0.119 − 0.262i)12-s + (0.232 − 1.61i)13-s + (0.224 − 0.144i)14-s + (−0.417 − 0.122i)15-s + (−0.0355 − 0.247i)16-s + (−0.984 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37759 - 0.265659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37759 - 0.265659i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.78 + 0.337i)T \) |
| good | 5 | \( 1 + (-1.41 - 0.910i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 4.24i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.837 + 5.82i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (4.05 + 4.68i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (0.509 - 0.587i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.83 - 4.42i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (4.41 + 1.29i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.57 + 2.93i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-10.2 - 6.58i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.57 + 1.04i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 0.443T + 47T^{2} \) |
| 53 | \( 1 + (0.184 + 1.28i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.938 + 6.52i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (8.45 + 2.48i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.983 + 2.15i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (0.452 + 0.991i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.618 - 0.714i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.80 - 12.5i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (5.89 - 3.79i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.47 + 1.31i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (4.42 + 2.84i)T + (40.2 + 88.2i)T^{2} \) |
| 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
−9.987247053561271141405638027157, −9.131063529213081751284887561069, −8.178993468040336283420817526375, −7.30305756244402224177930562388, −6.17016084994453262922941175954, −5.98574504202620671414456465883, −4.88123814943329960204809930262, −3.77923285447717937835194253938, −2.72412890835237167818809869872, −0.64490851090472743955532664944,
1.62819136720588532167642342817, 2.17532487604458078862358022240, 4.17400901078025792733944298545, 4.47691703276026472412932005000, 5.84982549015450077084683249818, 6.35275816169470677963569525102, 7.38624748214371445615006555998, 8.815950516634919331260590413955, 9.323259021336329075046166391360, 10.07702361386565044190585501285
