| L(s) = 1 | + (0.479 + 0.877i)2-s + (0.216 + 3.03i)3-s + (−0.540 + 0.841i)4-s + (1.67 − 1.47i)5-s + (−2.55 + 1.64i)6-s + (3.64 + 1.36i)7-s + (−0.997 − 0.0713i)8-s + (−6.17 + 0.887i)9-s + (2.10 + 0.761i)10-s + (−0.0570 − 0.194i)11-s + (−2.66 − 1.45i)12-s + (−2.31 − 6.20i)13-s + (0.553 + 3.85i)14-s + (4.84 + 4.75i)15-s + (−0.415 − 0.909i)16-s + (0.680 − 3.12i)17-s + ⋯ |
| L(s) = 1 | + (0.338 + 0.620i)2-s + (0.125 + 1.74i)3-s + (−0.270 + 0.420i)4-s + (0.749 − 0.661i)5-s + (−1.04 + 0.670i)6-s + (1.37 + 0.514i)7-s + (−0.352 − 0.0252i)8-s + (−2.05 + 0.295i)9-s + (0.664 + 0.240i)10-s + (−0.0172 − 0.0585i)11-s + (−0.769 − 0.420i)12-s + (−0.641 − 1.72i)13-s + (0.148 + 1.02i)14-s + (1.25 + 1.22i)15-s + (−0.103 − 0.227i)16-s + (0.165 − 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.856457 + 1.45199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.856457 + 1.45199i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.479 - 0.877i)T \) |
| 5 | \( 1 + (-1.67 + 1.47i)T \) |
| 23 | \( 1 + (3.15 - 3.61i)T \) |
| good | 3 | \( 1 + (-0.216 - 3.03i)T + (-2.96 + 0.426i)T^{2} \) |
| 7 | \( 1 + (-3.64 - 1.36i)T + (5.29 + 4.58i)T^{2} \) |
| 11 | \( 1 + (0.0570 + 0.194i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.31 + 6.20i)T + (-9.82 + 8.51i)T^{2} \) |
| 17 | \( 1 + (-0.680 + 3.12i)T + (-15.4 - 7.06i)T^{2} \) |
| 19 | \( 1 + (4.00 + 2.57i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.17 - 1.82i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.41 - 1.63i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.84 - 1.38i)T + (10.4 + 35.5i)T^{2} \) |
| 41 | \( 1 + (1.09 - 7.63i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.80 + 0.415i)T + (42.5 - 6.11i)T^{2} \) |
| 47 | \( 1 + (1.72 + 1.72i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.18 - 5.87i)T + (-40.0 - 34.7i)T^{2} \) |
| 59 | \( 1 + (2.26 + 1.03i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.61 + 1.39i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.80 + 0.984i)T + (36.2 - 56.3i)T^{2} \) |
| 71 | \( 1 + (11.8 + 3.46i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.02 + 0.441i)T + (66.4 - 30.3i)T^{2} \) |
| 79 | \( 1 + (5.83 - 12.7i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-3.77 + 5.04i)T + (-23.3 - 79.6i)T^{2} \) |
| 89 | \( 1 + (5.94 - 6.85i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (5.98 + 7.99i)T + (-27.3 + 93.0i)T^{2} \) |
| 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
−12.57665097258501307247710119972, −11.43540186313494866215183583708, −10.41883221458004416813693690651, −9.557300955240011257771438703429, −8.641954273289183091194828027990, −7.918269622177697752727269805072, −5.78393130164321911932516554538, −5.10260516519543227047756618387, −4.52471562363053496280857408948, −2.81001271248496425375573965958,
1.69709719583662724685010253655, 2.23689429426465799810275852876, 4.31750932476265943748238003748, 5.92760430137820935915248839828, 6.78641091719156035739587650291, 7.76179461501859795907070174745, 8.795428478691456079727510587045, 10.25189995163253452226092279800, 11.22795016447885958535393714240, 11.95515652722122414637319930486
