| L(s) = 1 | + (−0.550 + 1.30i)2-s + (−0.388 + 0.149i)3-s + (−1.39 − 1.43i)4-s + (−0.492 + 0.630i)5-s + (0.0187 − 0.589i)6-s + (1.24 − 3.47i)7-s + (2.63 − 1.02i)8-s + (−2.09 + 1.89i)9-s + (−0.550 − 0.988i)10-s + (0.585 + 2.60i)11-s + (0.757 + 0.348i)12-s + (−0.689 + 0.190i)13-s + (3.83 + 3.52i)14-s + (0.0968 − 0.319i)15-s + (−0.115 + 3.99i)16-s + (4.64 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.389 + 0.921i)2-s + (−0.224 + 0.0865i)3-s + (−0.696 − 0.717i)4-s + (−0.220 + 0.282i)5-s + (0.00764 − 0.240i)6-s + (0.469 − 1.31i)7-s + (0.931 − 0.362i)8-s + (−0.698 + 0.632i)9-s + (−0.174 − 0.312i)10-s + (0.176 + 0.785i)11-s + (0.218 + 0.100i)12-s + (−0.191 + 0.0529i)13-s + (1.02 + 0.943i)14-s + (0.0249 − 0.0823i)15-s + (−0.0288 + 0.999i)16-s + (1.12 − 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0209 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0209 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.682108 + 0.696553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.682108 + 0.696553i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.550 - 1.30i)T \) |
| good | 3 | \( 1 + (0.388 - 0.149i)T + (2.22 - 2.01i)T^{2} \) |
| 5 | \( 1 + (0.492 - 0.630i)T + (-1.21 - 4.85i)T^{2} \) |
| 7 | \( 1 + (-1.24 + 3.47i)T + (-5.41 - 4.44i)T^{2} \) |
| 11 | \( 1 + (-0.585 - 2.60i)T + (-9.94 + 4.70i)T^{2} \) |
| 13 | \( 1 + (0.689 - 0.190i)T + (11.1 - 6.68i)T^{2} \) |
| 17 | \( 1 + (-4.64 + 1.40i)T + (14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (-2.79 - 5.55i)T + (-11.3 + 15.2i)T^{2} \) |
| 23 | \( 1 + (-8.05 + 1.19i)T + (22.0 - 6.67i)T^{2} \) |
| 29 | \( 1 + (0.146 + 0.846i)T + (-27.3 + 9.76i)T^{2} \) |
| 31 | \( 1 + (3.05 - 4.57i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (5.10 - 4.40i)T + (5.42 - 36.5i)T^{2} \) |
| 41 | \( 1 + (-0.394 + 1.57i)T + (-36.1 - 19.3i)T^{2} \) |
| 43 | \( 1 + (-1.81 - 4.10i)T + (-28.8 + 31.8i)T^{2} \) |
| 47 | \( 1 + (-0.967 - 1.17i)T + (-9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (-1.41 + 8.16i)T + (-49.9 - 17.8i)T^{2} \) |
| 59 | \( 1 + (2.67 - 9.67i)T + (-50.6 - 30.3i)T^{2} \) |
| 61 | \( 1 + (-10.5 - 0.258i)T + (60.9 + 2.99i)T^{2} \) |
| 67 | \( 1 + (-2.36 + 2.25i)T + (3.28 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-5.79 + 0.284i)T + (70.6 - 6.95i)T^{2} \) |
| 73 | \( 1 + (4.88 + 13.6i)T + (-56.4 + 46.3i)T^{2} \) |
| 79 | \( 1 + (2.02 - 0.199i)T + (77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (-6.42 - 5.53i)T + (12.1 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.609 - 0.0904i)T + (85.1 + 25.8i)T^{2} \) |
| 97 | \( 1 + (15.1 - 3.00i)T + (89.6 - 37.1i)T^{2} \) |
| 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
−10.77834543424307536249009108042, −10.28482546964324993910855490452, −9.328200761508323653353262993499, −8.135574206446304040991267326429, −7.44829218727706720547754561857, −6.88455635177945989281394050214, −5.44817408014481305989248691883, −4.80756065826414069833238407554, −3.51505246625379010415856800458, −1.27226186790094104912751274442,
0.834049956578440528398753678656, 2.56507507620876251579173127937, 3.48132148722739059740303892853, 5.04036171967758547589170358335, 5.71715471467861290158664446686, 7.23748107747240451263123102836, 8.495401881210655359457980237694, 8.840559311333233732959003340471, 9.655060219991567550899273606527, 11.05633112658029865166821216314
