| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (3.67 + 2.12i)5-s − 2.82i·8-s + (−3 − 5.19i)10-s + (−11.0 + 6.36i)11-s + 13-s + (−2.00 + 3.46i)16-s + (14.6 − 8.48i)17-s + (11.5 − 19.9i)19-s + 8.48i·20-s + 18·22-s + (14.6 + 8.48i)23-s + (−3.5 − 6.06i)25-s + (−1.22 − 0.707i)26-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.734 + 0.424i)5-s − 0.353i·8-s + (−0.300 − 0.519i)10-s + (−1.00 + 0.578i)11-s + 0.0769·13-s + (−0.125 + 0.216i)16-s + (0.864 − 0.499i)17-s + (0.605 − 1.04i)19-s + 0.424i·20-s + 0.818·22-s + (0.638 + 0.368i)23-s + (−0.140 − 0.242i)25-s + (−0.0471 − 0.0271i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.498758783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498758783\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-3.67 - 2.12i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (11.0 - 6.36i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - T + 169T^{2} \) |
| 17 | \( 1 + (-14.6 + 8.48i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-11.5 + 19.9i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-14.6 - 8.48i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 33.9iT - 841T^{2} \) |
| 31 | \( 1 + (-23.5 - 40.7i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-27.5 + 47.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23T + 1.84e3T^{2} \) |
| 47 | \( 1 + (3.67 + 2.12i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (44.0 - 25.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (73.4 - 42.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-52 + 90.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-48.5 - 84.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 97.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-32.5 - 56.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (56.5 - 97.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 29.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-117. - 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 104T + 9.40e3T^{2} \) |
| 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
−9.890920030417361540936865427155, −9.451068398618163698043544787643, −8.403529008296025653251961079394, −7.43905607875893766092125162119, −6.83204225245243557456870946701, −5.61323856939952557296600190676, −4.76748319442098445423032070166, −3.14950856175465587134293012830, −2.46513254758780108580677343838, −1.07883334071412908599988494406,
0.72349979929513840154668891986, 1.99480493508842046158376806345, 3.27791016337674869306805119521, 4.79059161416349287125462381075, 5.78130473143812503631309748912, 6.18254520411938844040178923367, 7.70711218587262166814207742321, 8.022087726852077189581433658717, 9.083833636110077752395692940315, 9.865719310111557435783879479226
