| L(s) = 1 | + (0.109 − 0.994i)2-s + (2.20 − 0.323i)3-s + (−0.976 − 0.217i)4-s + (−2.19 + 0.0964i)5-s + (−0.0812 − 2.22i)6-s + (2.45 + 3.54i)7-s + (−0.322 + 0.946i)8-s + (1.86 − 0.562i)9-s + (−0.144 + 2.19i)10-s + (1.87 + 2.88i)11-s + (−2.21 − 0.162i)12-s + (1.27 + 1.40i)13-s + (3.79 − 2.05i)14-s + (−4.81 + 0.925i)15-s + (0.905 + 0.424i)16-s + (0.0570 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.0773 − 0.702i)2-s + (1.27 − 0.187i)3-s + (−0.488 − 0.108i)4-s + (−0.983 + 0.0431i)5-s + (−0.0331 − 0.907i)6-s + (0.928 + 1.33i)7-s + (−0.114 + 0.334i)8-s + (0.622 − 0.187i)9-s + (−0.0457 + 0.694i)10-s + (0.566 + 0.870i)11-s + (−0.640 − 0.0468i)12-s + (0.354 + 0.390i)13-s + (1.01 − 0.549i)14-s + (−1.24 + 0.238i)15-s + (0.226 + 0.106i)16-s + (0.0138 − 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.18184 - 0.0225836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18184 - 0.0225836i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.109 + 0.994i)T \) |
| 431 | \( 1 + (-13.7 + 15.5i)T \) |
| good | 3 | \( 1 + (-2.20 + 0.323i)T + (2.87 - 0.864i)T^{2} \) |
| 5 | \( 1 + (2.19 - 0.0964i)T + (4.98 - 0.437i)T^{2} \) |
| 7 | \( 1 + (-2.45 - 3.54i)T + (-2.45 + 6.55i)T^{2} \) |
| 11 | \( 1 + (-1.87 - 2.88i)T + (-4.44 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 1.40i)T + (-1.23 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0570 + 1.11i)T + (-16.9 - 1.73i)T^{2} \) |
| 19 | \( 1 + (-0.412 - 1.48i)T + (-16.2 + 9.77i)T^{2} \) |
| 23 | \( 1 + (-0.609 - 0.145i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-3.86 - 1.67i)T + (19.8 + 21.1i)T^{2} \) |
| 31 | \( 1 + (0.248 + 0.0907i)T + (23.6 + 20.0i)T^{2} \) |
| 37 | \( 1 + (-1.81 - 1.72i)T + (1.89 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-1.38 + 1.47i)T + (-2.69 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.250 + 3.80i)T + (-42.6 + 5.63i)T^{2} \) |
| 47 | \( 1 + (-7.35 + 10.2i)T + (-15.1 - 44.4i)T^{2} \) |
| 53 | \( 1 + (-3.48 + 4.72i)T + (-15.6 - 50.6i)T^{2} \) |
| 59 | \( 1 + (-2.74 - 1.23i)T + (39.0 + 44.2i)T^{2} \) |
| 61 | \( 1 + (-3.05 + 2.81i)T + (4.89 - 60.8i)T^{2} \) |
| 67 | \( 1 + (1.33 - 2.80i)T + (-42.1 - 52.1i)T^{2} \) |
| 71 | \( 1 + (0.0866 + 11.8i)T + (-70.9 + 1.03i)T^{2} \) |
| 73 | \( 1 + (3.62 - 2.32i)T + (30.5 - 66.3i)T^{2} \) |
| 79 | \( 1 + (10.4 - 0.304i)T + (78.8 - 4.61i)T^{2} \) |
| 83 | \( 1 + (11.9 + 6.25i)T + (47.3 + 68.2i)T^{2} \) |
| 89 | \( 1 + (4.05 - 1.97i)T + (54.9 - 70.0i)T^{2} \) |
| 97 | \( 1 + (-3.31 - 11.3i)T + (-81.6 + 52.3i)T^{2} \) |
| 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
−10.03687319752332582494709177188, −9.048298009036441023443478585141, −8.636142767310613746454519791740, −7.945803548304898950797989760164, −7.05539652791007699162961321122, −5.54395371024328923137812986308, −4.45328039565324322943335318954, −3.61501420949203624633479029325, −2.51928379303508200857770383333, −1.70711338015244271834232614194,
1.00302175590974167228733510604, 3.02041731811293387111646769449, 4.00943454782589983993893532411, 4.37223103737386740155331964742, 5.87452623465266255596035306259, 7.13290597127850068539958232179, 7.78317094878769698629009082520, 8.307923238028414882760125368184, 8.916504197213773294515848922469, 10.01635517598144364027225620201
