L(s) = 1 | + (−3.43 + 1.98i)5-s − 2.01i·7-s − 3.45·11-s + (0.581 − 1.00i)13-s + (1.94 − 1.12i)17-s + (−4.27 − 0.843i)19-s + (−1.36 + 2.36i)23-s + (5.36 − 9.28i)25-s + (0.0525 + 0.0303i)29-s − 2.92i·31-s + (3.99 + 6.91i)35-s + 10.5·37-s + (−9.71 + 5.60i)41-s + (−6.21 + 3.58i)43-s + (5.99 − 10.3i)47-s + ⋯ |
L(s) = 1 | + (−1.53 + 0.886i)5-s − 0.761i·7-s − 1.04·11-s + (0.161 − 0.279i)13-s + (0.470 − 0.271i)17-s + (−0.981 − 0.193i)19-s + (−0.284 + 0.493i)23-s + (1.07 − 1.85i)25-s + (0.00976 + 0.00563i)29-s − 0.525i·31-s + (0.674 + 1.16i)35-s + 1.73·37-s + (−1.51 + 0.875i)41-s + (−0.947 + 0.547i)43-s + (0.873 − 1.51i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8726980107\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8726980107\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4.27 + 0.843i)T \) |
good | 5 | \( 1 + (3.43 - 1.98i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 2.01iT - 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 + (-0.581 + 1.00i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.36 - 2.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0525 - 0.0303i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.92iT - 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + (9.71 - 5.60i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.21 - 3.58i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.99 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.31 - 1.33i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.08 - 10.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.76 - 6.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.77 - 1.60i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.20 + 7.29i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.71 - 2.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.16 - 2.98i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + (-12.2 - 7.04i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.03 - 8.72i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.572261864689284867800590562388, −7.980874701191139072854001055243, −7.45093677247437159668320244762, −6.86536073007172395710537200897, −5.90618512138810950609884225011, −4.77578897623652532058511027674, −4.02457388164523391724878397681, −3.32181037401178591594718291621, −2.45782147449965002808747950834, −0.64370446966638492358174896396,
0.49027882391612058822055414169, 2.00548641234453205772781092252, 3.16884073253489620814408839116, 4.06261196655866960924162095415, 4.79042026590506436114345592054, 5.50572610223621680225529872112, 6.47917636386443364470226092661, 7.50132305412264357929708946519, 8.165433698406427453014646645196, 8.518236271983790319451886207936