L(s) = 1 | + (−0.151 + 1.40i)2-s + 2.99i·3-s + (−1.95 − 0.425i)4-s + i·5-s + (−4.21 − 0.453i)6-s + 3.98·7-s + (0.893 − 2.68i)8-s − 5.98·9-s + (−1.40 − 0.151i)10-s − 5.79·11-s + (1.27 − 5.85i)12-s − 5.59·13-s + (−0.602 + 5.59i)14-s − 2.99·15-s + (3.63 + 1.66i)16-s + 1.88i·17-s + ⋯ |
L(s) = 1 | + (−0.106 + 0.994i)2-s + 1.73i·3-s + (−0.977 − 0.212i)4-s + 0.447i·5-s + (−1.72 − 0.185i)6-s + 1.50·7-s + (0.315 − 0.948i)8-s − 1.99·9-s + (−0.444 − 0.0478i)10-s − 1.74·11-s + (0.368 − 1.69i)12-s − 1.55·13-s + (−0.160 + 1.49i)14-s − 0.774·15-s + (0.909 + 0.415i)16-s + 0.456i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.441970 - 0.850728i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.441970 - 0.850728i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.151 - 1.40i)T \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-4.42 - 1.85i)T \) |
good | 3 | \( 1 - 2.99iT - 3T^{2} \) |
| 7 | \( 1 - 3.98T + 7T^{2} \) |
| 11 | \( 1 + 5.79T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 - 1.88iT - 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 4.43iT - 31T^{2} \) |
| 37 | \( 1 + 4.01iT - 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 - 6.45iT - 47T^{2} \) |
| 53 | \( 1 - 8.62iT - 53T^{2} \) |
| 59 | \( 1 - 1.55iT - 59T^{2} \) |
| 61 | \( 1 - 6.06iT - 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 - 5.26iT - 71T^{2} \) |
| 73 | \( 1 - 2.20T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8.56T + 83T^{2} \) |
| 89 | \( 1 + 8.41iT - 89T^{2} \) |
| 97 | \( 1 + 2.87iT - 97T^{2} \) |
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
−11.15746801067043255971326232320, −10.52997501818797086249122821872, −9.884112496296329222903266765096, −8.941614086313297277975269415817, −7.971609101245998314781228495460, −7.34462840742329801187190302124, −5.47510199844017187060771949450, −5.10999497199785456639401690401, −4.35438417574474404048069356418, −2.85512985426219632672877783302,
0.61214061972460443618340604014, 1.99565578732841433138515620392, 2.70009590510981186090186997322, 4.96886432780788319119972028431, 5.25483354048829077739661550086, 7.19768456071297107840759877993, 7.929457395327154634284168809856, 8.311876130051242789514409996929, 9.621551956258333915691909472452, 10.78954446804749818840699349094