| L(s) = 1 | + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−5.31 − 3.06i)5-s + (5.10 + 4.78i)7-s + 2.82i·8-s + (−4.33 − 7.51i)10-s + (8.42 − 4.86i)11-s + 21.0·13-s + (2.86 + 9.47i)14-s + (−2.00 + 3.46i)16-s + (−2.48 + 1.43i)17-s + (−18.4 + 32.0i)19-s − 12.2i·20-s + 13.7·22-s + (38.2 + 22.1i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.06 − 0.613i)5-s + (0.729 + 0.683i)7-s + 0.353i·8-s + (−0.433 − 0.751i)10-s + (0.765 − 0.442i)11-s + 1.62·13-s + (0.204 + 0.676i)14-s + (−0.125 + 0.216i)16-s + (−0.146 + 0.0843i)17-s + (−0.973 + 1.68i)19-s − 0.613i·20-s + 0.625·22-s + (1.66 + 0.961i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.09533 + 0.987764i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09533 + 0.987764i\) |
| \(L(2)\) |
|
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 | \( 1 + (-5.10 - 4.78i)T \) |
| good | 5 | \( 1 + (5.31 + 3.06i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-8.42 + 4.86i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 21.0T + 169T^{2} \) |
| 17 | \( 1 + (2.48 - 1.43i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (18.4 - 32.0i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-38.2 - 22.1i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 7.81iT - 841T^{2} \) |
| 31 | \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-11.9 + 20.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 49.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 58.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + (58.3 + 33.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-28.4 + 16.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (61.5 - 35.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.3 + 40.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (9.40 + 16.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 27.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (43.8 + 76.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (39.8 - 69.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 93.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (15.2 + 8.83i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 17.2T + 9.40e3T^{2} \) |
| 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
−11.43639521811267519971246240216, −10.79860741179610637956174403490, −8.877630387479411215544315853149, −8.547822617369294254308093563452, −7.62563372585751466386342483355, −6.31426910008615514705233684594, −5.43554101396261551703283905763, −4.22400871632052589200380800137, −3.49075505190961714025562000615, −1.43581832140213710697563453123,
1.04372092479908806108076163086, 2.87098293640297099478381240564, 4.10388952759822225512673098316, 4.62382564585337767131886708427, 6.40267060665327042989199640353, 7.03501113652322586673594878684, 8.167731123867600790742901183109, 9.147498169238363085838347068500, 10.65467227307644366444316793028, 11.18890925383177888039493481131
