| L(s) = 1 | + (0.900 − 0.433i)2-s + (−1.43 − 0.972i)3-s + (0.623 − 0.781i)4-s + (1.92 + 1.78i)5-s + (−1.71 − 0.254i)6-s + (2.34 − 1.21i)7-s + (0.222 − 0.974i)8-s + (1.10 + 2.78i)9-s + (2.50 + 0.773i)10-s + (−0.0164 − 0.0112i)11-s + (−1.65 + 0.513i)12-s + (−1.93 − 1.31i)13-s + (1.58 − 2.11i)14-s + (−1.02 − 4.43i)15-s + (−0.222 − 0.974i)16-s + (3.23 − 0.487i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.306i)2-s + (−0.827 − 0.561i)3-s + (0.311 − 0.390i)4-s + (0.860 + 0.798i)5-s + (−0.699 − 0.104i)6-s + (0.888 − 0.459i)7-s + (0.0786 − 0.344i)8-s + (0.368 + 0.929i)9-s + (0.793 + 0.244i)10-s + (−0.00496 − 0.00338i)11-s + (−0.477 + 0.148i)12-s + (−0.536 − 0.366i)13-s + (0.424 − 0.565i)14-s + (−0.263 − 1.14i)15-s + (−0.0556 − 0.243i)16-s + (0.784 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.13036 - 0.778709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13036 - 0.778709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 3 | \( 1 + (1.43 + 0.972i)T \) |
| 7 | \( 1 + (-2.34 + 1.21i)T \) |
| good | 5 | \( 1 + (-1.92 - 1.78i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (0.0164 + 0.0112i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (1.93 + 1.31i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-3.23 + 0.487i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-3.63 - 6.30i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.04 - 2.67i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (6.31 - 0.952i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 - 9.78T + 31T^{2} \) |
| 37 | \( 1 + (-3.70 + 9.44i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (9.91 - 3.05i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (2.10 + 0.647i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-7.99 + 3.84i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (3.01 + 7.67i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-1.85 - 8.14i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (5.99 + 7.51i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + (-3.90 + 4.89i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.463 + 0.316i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 - 5.25T + 79T^{2} \) |
| 83 | \( 1 + (-9.17 + 6.25i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-1.24 - 16.6i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (6.79 - 11.7i)T + (-48.5 - 84.0i)T^{2} \) |
| 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
−10.32860294560432980616464976527, −9.676679485449772607109919178423, −7.916821512778306656890490250441, −7.40105364318290441746017911461, −6.39098556878540739436637457186, −5.59072618015193611787935300254, −5.00848494321723127288896765271, −3.63023892353926183293843716329, −2.29324156880089230257102177389, −1.29261998753910624285934237697,
1.32577421410798025112437305845, 2.84843384942385957571838757550, 4.47275111729877914574386788527, 4.97132880921794421776942688754, 5.59526800810897806398043782698, 6.47401295805526992567067969983, 7.53704961785216239222090468108, 8.692073241098975548286464836008, 9.419912312349569887469256842849, 10.20468122760652665382510150761
