| L(s) = 1 | + (1.18 + 1.83i)2-s + (−3.21 − 0.462i)3-s + (−1.15 + 2.52i)4-s + (−1.36 + 1.77i)5-s + (−2.95 − 6.46i)6-s + (−1.56 + 1.35i)7-s + (−1.68 + 0.242i)8-s + (7.27 + 2.13i)9-s + (−4.87 − 0.416i)10-s + (−0.477 − 0.306i)11-s + (4.88 − 7.60i)12-s + (1.57 + 1.36i)13-s + (−4.35 − 1.27i)14-s + (5.21 − 5.07i)15-s + (1.20 + 1.38i)16-s + (1.77 − 0.812i)17-s + ⋯ |
| L(s) = 1 | + (0.835 + 1.30i)2-s + (−1.85 − 0.267i)3-s + (−0.577 + 1.26i)4-s + (−0.610 + 0.792i)5-s + (−1.20 − 2.64i)6-s + (−0.593 + 0.513i)7-s + (−0.596 + 0.0857i)8-s + (2.42 + 0.711i)9-s + (−1.54 − 0.131i)10-s + (−0.143 − 0.0924i)11-s + (1.41 − 2.19i)12-s + (0.437 + 0.378i)13-s + (−1.16 − 0.341i)14-s + (1.34 − 1.30i)15-s + (0.300 + 0.346i)16-s + (0.431 − 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0907518 + 0.731246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0907518 + 0.731246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.36 - 1.77i)T \) |
| 23 | \( 1 + (-3.06 + 3.69i)T \) |
| good | 2 | \( 1 + (-1.18 - 1.83i)T + (-0.830 + 1.81i)T^{2} \) |
| 3 | \( 1 + (3.21 + 0.462i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (1.56 - 1.35i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (0.477 + 0.306i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.57 - 1.36i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.77 + 0.812i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (2.66 - 5.84i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.622 + 1.36i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.135 - 0.942i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.57 - 5.36i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.72 + 0.801i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-5.10 - 0.734i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 4.83iT - 47T^{2} \) |
| 53 | \( 1 + (-0.0581 + 0.0503i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-8.80 + 10.1i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.16 + 8.13i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.26 - 8.19i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (3.53 - 2.27i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-11.1 - 5.08i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.68 + 4.25i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (1.03 - 3.52i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.18 - 8.20i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.75 + 5.97i)T + (-81.6 + 52.4i)T^{2} \) |
| 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
−14.19758101101566562600837670007, −12.85799744165225763413439109808, −12.24580405767616058060441837632, −11.16816062371728229804078364569, −10.19790220986445804932018425537, −8.000394673019137097005964718604, −6.79481811605950697022577013665, −6.30449045243191526664840461663, −5.33896329757262770939951991017, −4.00512194245424736239466070100,
0.830851717048720665656053745483, 3.78230074988641750290277002286, 4.74278450938210258598067948406, 5.69306016102871076529073949680, 7.22955311232332175654677758792, 9.443835952130702807680739495813, 10.61743027931266090434828640732, 11.13645924516833373557724739452, 12.03581504959492490673447645494, 12.81507944923639020086316432722
