| L(s) = 1 | + (0.816 + 2.24i)2-s + (0.939 + 0.342i)3-s + (−2.83 + 2.38i)4-s + (0.143 − 0.0252i)5-s + 2.38i·6-s + (−0.799 − 4.53i)7-s + (−3.52 − 2.03i)8-s + (0.766 + 0.642i)9-s + (0.173 + 0.300i)10-s + (−0.876 + 1.51i)11-s + (−3.48 + 1.26i)12-s + (−1.50 − 1.79i)13-s + (9.52 − 5.49i)14-s + (0.143 + 0.0252i)15-s + (0.401 − 2.27i)16-s + (2.96 − 3.53i)17-s + ⋯ |
| L(s) = 1 | + (0.577 + 1.58i)2-s + (0.542 + 0.197i)3-s + (−1.41 + 1.19i)4-s + (0.0640 − 0.0112i)5-s + 0.975i·6-s + (−0.302 − 1.71i)7-s + (−1.24 − 0.719i)8-s + (0.255 + 0.214i)9-s + (0.0549 + 0.0951i)10-s + (−0.264 + 0.457i)11-s + (−1.00 + 0.365i)12-s + (−0.416 − 0.496i)13-s + (2.54 − 1.46i)14-s + (0.0369 + 0.00651i)15-s + (0.100 − 0.569i)16-s + (0.719 − 0.857i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.824765 + 1.17255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.824765 + 1.17255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 37 | \( 1 + (-5.25 + 3.06i)T \) |
| good | 2 | \( 1 + (-0.816 - 2.24i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.143 + 0.0252i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.799 + 4.53i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.876 - 1.51i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.50 + 1.79i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.96 + 3.53i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (2.17 - 5.97i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (1.27 - 0.734i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.93 - 4.00i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.30iT - 31T^{2} \) |
| 41 | \( 1 + (1.69 - 1.42i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 1.21iT - 43T^{2} \) |
| 47 | \( 1 + (0.780 + 1.35i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.07 - 11.7i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (11.6 + 2.06i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.44 + 1.72i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (0.803 + 4.55i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.54 + 1.65i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + (0.536 - 0.0946i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.97 - 5.01i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-13.8 - 2.44i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.07 + 5.24i)T + (48.5 - 84.0i)T^{2} \) |
| 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
−14.06622930062721844914411684138, −13.50021419893828701434481500842, −12.46294061576104099578887291416, −10.47967067151240833410254388724, −9.575191183016237239659043589410, −7.76747076845009940675632532182, −7.60136103281479786464640457576, −6.20316253945738273299708277356, −4.73081973819497843528773138093, −3.67674275669780482563216188422,
2.17020413850756102035423361539, 3.12528015074248731978494067424, 4.79345027544278003814721217324, 6.19876200933188309043890698781, 8.367853228707158098220208436576, 9.291659120482665952262861037414, 10.26688269088120869954046711525, 11.60111143612290430528098550747, 12.25670281950989353670400252328, 13.04429802392936278607287005901
