| L(s) = 1 | + (−0.451 + 1.34i)2-s + (0.924 + 2.85i)3-s + (−1.59 − 1.21i)4-s + (−0.606 − 0.241i)5-s + (−4.24 − 0.0499i)6-s + (6.70 − 3.10i)7-s + (2.34 − 1.58i)8-s + (−7.29 + 5.27i)9-s + (0.597 − 0.703i)10-s + (−16.9 − 4.69i)11-s + (1.98 − 5.66i)12-s + (−8.35 + 15.7i)13-s + (1.12 + 10.3i)14-s + (0.128 − 1.95i)15-s + (1.07 + 3.85i)16-s + (−7.23 + 15.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.670i)2-s + (0.308 + 0.951i)3-s + (−0.398 − 0.302i)4-s + (−0.121 − 0.0482i)5-s + (−0.707 − 0.00832i)6-s + (0.958 − 0.443i)7-s + (0.292 − 0.198i)8-s + (−0.810 + 0.586i)9-s + (0.0597 − 0.0703i)10-s + (−1.53 − 0.427i)11-s + (0.165 − 0.471i)12-s + (−0.642 + 1.21i)13-s + (0.0807 + 0.742i)14-s + (0.00859 − 0.130i)15-s + (0.0668 + 0.240i)16-s + (−0.425 + 0.919i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.185910 - 0.683300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185910 - 0.683300i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (-0.924 - 2.85i)T \) |
| 59 | \( 1 + (47.3 - 35.1i)T \) |
| good | 5 | \( 1 + (0.606 + 0.241i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-6.70 + 3.10i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (16.9 + 4.69i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (8.35 - 15.7i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (7.23 - 15.6i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (23.1 - 5.08i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-2.68 - 0.439i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-9.66 - 28.6i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (-11.7 - 2.57i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-13.2 + 19.5i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (26.4 - 4.33i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-5.94 - 21.3i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-64.0 + 25.5i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-48.0 + 40.8i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-33.4 - 11.2i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-50.8 - 75.0i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-49.3 + 19.6i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (96.5 - 10.5i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (102. - 61.7i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (59.2 - 3.21i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (1.85 + 5.50i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (121. + 13.2i)T + (9.18e3 + 2.02e3i)T^{2} \) |
| 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
−11.44886126224110464993176825595, −10.60887901087596814474668206262, −10.05620841432695176227774718831, −8.673183104705658962968234436722, −8.298784720682833701190588296286, −7.25001634054984795292203943573, −5.84978533248370188148870568804, −4.77239872915864838588634557314, −4.10504919434089211022773126535, −2.25515529143895728904222360282,
0.31101387982844287542547134185, 2.14451887625892454277536069833, 2.80886234427664962101712667429, 4.69390777785567849254286492521, 5.67639066866796456546967453122, 7.30921004579261261669722324166, 7.950078145364401000500470062561, 8.642682187220216344643494083091, 9.885811694630101063645472081655, 10.85058196822053898095848593006
