| 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 | \( 1 + (5.98 - 7.99i)T + (-27.3 - 93.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95515652722122414637319930486, −11.22795016447885958535393714240, −10.25189995163253452226092279800, −8.795428478691456079727510587045, −7.76179461501859795907070174745, −6.78641091719156035739587650291, −5.92760430137820935915248839828, −4.31750932476265943748238003748, −2.23689429426465799810275852876, −1.69709719583662724685010253655,
2.81001271248496425375573965958, 4.52471562363053496280857408948, 5.10260516519543227047756618387, 5.78393130164321911932516554538, 7.918269622177697752727269805072, 8.641954273289183091194828027990, 9.557300955240011257771438703429, 10.41883221458004416813693690651, 11.43540186313494866215183583708, 12.57665097258501307247710119972
