L(s) = 1 | + (−9.12 + 15.7i)5-s + (−9.48 − 15.9i)7-s + (49.3 − 28.4i)11-s + (9.36 + 5.40i)13-s − 65.2·17-s + 36.6i·19-s + (70.2 + 40.5i)23-s + (−103. − 179. i)25-s + (233. − 134. i)29-s + (−117. − 67.9i)31-s + (337. − 4.68i)35-s − 125.·37-s + (117. − 203. i)41-s + (22.6 + 39.2i)43-s + (241. + 417. i)47-s + ⋯ |
L(s) = 1 | + (−0.815 + 1.41i)5-s + (−0.511 − 0.859i)7-s + (1.35 − 0.780i)11-s + (0.199 + 0.115i)13-s − 0.931·17-s + 0.441i·19-s + (0.636 + 0.367i)23-s + (−0.830 − 1.43i)25-s + (1.49 − 0.863i)29-s + (−0.682 − 0.393i)31-s + (1.63 − 0.0226i)35-s − 0.558·37-s + (0.447 − 0.775i)41-s + (0.0802 + 0.139i)43-s + (0.748 + 1.29i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.464571635\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.464571635\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (9.48 + 15.9i)T \) |
good | 5 | \( 1 + (9.12 - 15.7i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-49.3 + 28.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-9.36 - 5.40i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 65.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-70.2 - 40.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-233. + 134. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (117. + 67.9i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-117. + 203. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-22.6 - 39.2i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-241. - 417. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 70.1iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (176. - 306. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-512. + 295. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (261. - 453. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 895. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 982. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-510. - 883. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (152. + 263. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (677. - 391. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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
−10.27023054533377002798231999839, −9.294700898142367985319323713334, −8.314727521156819305172326172275, −7.25664404037864658110486227625, −6.73815778880913528338363978703, −6.02505931137292042143843565753, −4.17708737888445982337828793971, −3.71438884083873781548624870366, −2.69841646632579656632822679023, −0.890872153309908023395921106632,
0.54559703859382216540239259833, 1.80707984534524228572343297268, 3.35824520387959619435449274987, 4.46421964465080399614926932788, 5.04909439973932036366499912120, 6.35674400971981334410467921652, 7.12554217112191925879068151984, 8.423345050371605075503007573664, 8.940263589532344870146257288084, 9.396505062935303381233960000874