| L(s) = 1 | + (0.240 + 0.222i)3-s + (0.0260 − 0.00392i)5-s + (−1.56 − 2.71i)7-s + (−0.216 − 2.88i)9-s + (−3.36 − 1.62i)11-s + (−1.78 + 4.55i)13-s + (0.00712 + 0.00485i)15-s + (−5.84 − 0.881i)17-s + (−0.296 + 3.95i)19-s + (0.228 − 0.999i)21-s + (−0.284 + 0.193i)23-s + (−4.77 + 1.47i)25-s + (1.20 − 1.50i)27-s + (0.971 − 0.901i)29-s + (−2.06 − 0.638i)31-s + ⋯ |
| L(s) = 1 | + (0.138 + 0.128i)3-s + (0.0116 − 0.00175i)5-s + (−0.591 − 1.02i)7-s + (−0.0720 − 0.961i)9-s + (−1.01 − 0.488i)11-s + (−0.496 + 1.26i)13-s + (0.00183 + 0.00125i)15-s + (−1.41 − 0.213i)17-s + (−0.0680 + 0.908i)19-s + (0.0497 − 0.218i)21-s + (−0.0592 + 0.0403i)23-s + (−0.955 + 0.294i)25-s + (0.231 − 0.290i)27-s + (0.180 − 0.167i)29-s + (−0.371 − 0.114i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.124995 - 0.487108i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124995 - 0.487108i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (6.54 + 0.341i)T \) |
| good | 3 | \( 1 + (-0.240 - 0.222i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.0260 + 0.00392i)T + (4.77 - 1.47i)T^{2} \) |
| 7 | \( 1 + (1.56 + 2.71i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.36 + 1.62i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (1.78 - 4.55i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (5.84 + 0.881i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (0.296 - 3.95i)T + (-18.7 - 2.83i)T^{2} \) |
| 23 | \( 1 + (0.284 - 0.193i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.971 + 0.901i)T + (2.16 - 28.9i)T^{2} \) |
| 31 | \( 1 + (2.06 + 0.638i)T + (25.6 + 17.4i)T^{2} \) |
| 37 | \( 1 + (-5.01 + 8.69i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.431 + 1.89i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (-10.0 + 4.82i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (2.12 + 5.41i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (0.107 - 0.134i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-5.31 + 1.63i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.0822 + 1.09i)T + (-66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-4.37 - 2.98i)T + (25.9 + 66.0i)T^{2} \) |
| 73 | \( 1 + (2.70 - 6.88i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (2.70 + 4.68i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.49 - 1.38i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (-2.93 - 2.72i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (8.11 + 3.90i)T + (60.4 + 75.8i)T^{2} \) |
| 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
−9.999005248965550292576164507551, −9.384683762407456604377860095785, −8.467118694715743841003048030873, −7.33138661672249920890461934313, −6.68164550621375887160698343709, −5.68861595690900114433110995949, −4.28478253194753704679958891298, −3.64527825140052756831655721586, −2.22464336726275088892102931486, −0.23651659774160190523112134273,
2.31080148875647630015800492538, 2.83662100587649897865542263548, 4.62349696712084983404312914219, 5.37287052838529286522140920395, 6.34964185979748359804590939548, 7.49549170358329224596681908911, 8.200709031808928205577800749420, 9.066135521025908360746658093302, 10.04600874430236410368373907592, 10.71002293212129667712021005116
