L(s) = 1 | + (−2.59 − 4.50i)3-s + (−8.05 − 13.9i)5-s + (5.67 + 17.6i)7-s + (−13.5 + 23.3i)9-s + (−30.8 − 17.7i)11-s − 7.40i·13-s + (−41.9 + 72.4i)15-s + (−14.4 + 25.0i)17-s + (−30.4 + 17.5i)19-s + (64.6 − 71.2i)21-s + (−48.0 + 27.7i)23-s + (−67.3 + 116. i)25-s + (140. + 0.420i)27-s − 68.1i·29-s + (154. + 89.3i)31-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)3-s + (−0.720 − 1.24i)5-s + (0.306 + 0.951i)7-s + (−0.501 + 0.865i)9-s + (−0.845 − 0.487i)11-s − 0.158i·13-s + (−0.722 + 1.24i)15-s + (−0.206 + 0.357i)17-s + (−0.367 + 0.212i)19-s + (0.671 − 0.740i)21-s + (−0.435 + 0.251i)23-s + (−0.539 + 0.933i)25-s + (0.999 + 0.00299i)27-s − 0.436i·29-s + (0.896 + 0.517i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5437027457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5437027457\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.59 + 4.50i)T \) |
| 7 | \( 1 + (-5.67 - 17.6i)T \) |
good | 5 | \( 1 + (8.05 + 13.9i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (30.8 + 17.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 7.40iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (14.4 - 25.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (30.4 - 17.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (48.0 - 27.7i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 68.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-154. - 89.3i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-116. - 202. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 370.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-87.3 - 151. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (235. + 136. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-48.4 + 83.8i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (333. - 192. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (509. - 881. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 125. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-195. - 112. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (532. + 921. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 601.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-752. - 1.30e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 327. iT - 9.12e5T^{2} \) |
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
−11.57369170985079003161599040166, −10.59446691118417452315217830460, −9.094488456862205311867572771345, −8.179480859948325930206712942132, −7.84352123323279613344374986946, −6.22258868273828360190917939001, −5.40929946866665191575599510418, −4.46515249260122874435137511738, −2.57680342924427340015158676599, −1.10741071718529465301268641954,
0.23946650388903427875713569665, 2.72260839566187879189768621720, 3.93932131854150714319198897901, 4.70711561491265161870557585331, 6.15843957277302318441424478450, 7.17420137554534778179175768471, 7.922718558539933115898095616571, 9.387608267902169869115791912616, 10.43340281787721103595026281928, 10.81672337969843148777431261605