| L(s) = 1 | + (0.347 − 1.96i)2-s + (−1.62 − 9.18i)3-s + (−3.75 − 1.36i)4-s + (−9.08 + 7.62i)5-s − 18.6·6-s + (21.4 − 17.9i)7-s + (−4 + 6.92i)8-s + (−56.4 + 20.5i)9-s + (11.8 + 20.5i)10-s + (−4.89 + 8.47i)11-s + (−6.48 + 36.7i)12-s + (20.4 + 7.44i)13-s + (−27.9 − 48.4i)14-s + (84.8 + 71.1i)15-s + (12.2 + 10.2i)16-s + (−51.5 + 18.7i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.311 − 1.76i)3-s + (−0.469 − 0.171i)4-s + (−0.812 + 0.682i)5-s − 1.26·6-s + (1.15 − 0.970i)7-s + (−0.176 + 0.306i)8-s + (−2.09 + 0.760i)9-s + (0.375 + 0.649i)10-s + (−0.134 + 0.232i)11-s + (−0.155 + 0.884i)12-s + (0.436 + 0.158i)13-s + (−0.533 − 0.924i)14-s + (1.45 + 1.22i)15-s + (0.191 + 0.160i)16-s + (−0.735 + 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.173551 + 0.992304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.173551 + 0.992304i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 + 1.96i)T \) |
| 37 | \( 1 + (151. + 166. i)T \) |
| good | 3 | \( 1 + (1.62 + 9.18i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (9.08 - 7.62i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-21.4 + 17.9i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (4.89 - 8.47i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-20.4 - 7.44i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (51.5 - 18.7i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 19 | \( 1 + (28.0 + 158. i)T + (-6.44e3 + 2.34e3i)T^{2} \) |
| 23 | \( 1 + (56.1 + 97.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-90.1 + 156. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 162.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-296. - 108. i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-65.0 - 112. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-560. - 470. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (233. + 196. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (86.0 + 31.3i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-493. + 414. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (78.6 + 446. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 - 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (280. - 235. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (536. - 195. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (146. + 122. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-533. - 923. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
| 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
−13.44670828161821874542235925582, −12.26828522350773468511091722706, −11.24446511031965115219102348100, −10.88819967067431848655106499717, −8.451509299654627124870838220490, −7.50924445806410200871621384760, −6.55363627952185630677222402607, −4.47347836275807085872287684980, −2.34234727424754358195256262828, −0.66939797199179323198815930766,
3.84409631913278110950347874636, 4.87842566819076648977937786850, 5.75555754662425305029678894551, 8.273946966594177071750186314013, 8.682070670761196941490178480133, 10.13286353426086032931005490151, 11.40973587576188800230694243484, 12.18313220652416092762477792947, 14.13223049123138220950056698631, 15.05896989564876804147171516942
