L(s) = 1 | + (1.39 − 0.202i)2-s + (−0.0300 − 1.73i)3-s + (1.91 − 0.567i)4-s + (0.5 + 0.866i)5-s + (−0.392 − 2.41i)6-s + (3.45 + 1.99i)7-s + (2.56 − 1.18i)8-s + (−2.99 + 0.104i)9-s + (0.875 + 1.11i)10-s + (0.520 + 0.300i)11-s + (−1.03 − 3.30i)12-s + (−4.44 + 2.56i)13-s + (5.23 + 2.08i)14-s + (1.48 − 0.891i)15-s + (3.35 − 2.17i)16-s − 5.37i·17-s + ⋯ |
L(s) = 1 | + (0.989 − 0.143i)2-s + (−0.0173 − 0.999i)3-s + (0.958 − 0.283i)4-s + (0.223 + 0.387i)5-s + (−0.160 − 0.987i)6-s + (1.30 + 0.753i)7-s + (0.908 − 0.417i)8-s + (−0.999 + 0.0347i)9-s + (0.276 + 0.351i)10-s + (0.157 + 0.0906i)11-s + (−0.300 − 0.953i)12-s + (−1.23 + 0.711i)13-s + (1.39 + 0.558i)14-s + (0.383 − 0.230i)15-s + (0.839 − 0.543i)16-s − 1.30i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.40352 - 0.954192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.40352 - 0.954192i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 + 0.202i)T \) |
| 3 | \( 1 + (0.0300 + 1.73i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-3.45 - 1.99i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.520 - 0.300i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.44 - 2.56i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.37iT - 17T^{2} \) |
| 19 | \( 1 + 8.29T + 19T^{2} \) |
| 23 | \( 1 + (-0.0667 - 0.115i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.507 + 0.879i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.58 - 0.914i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.87iT - 37T^{2} \) |
| 41 | \( 1 + (-4.43 + 2.56i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.36 - 5.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.27 - 9.13i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6.25T + 53T^{2} \) |
| 59 | \( 1 + (-10.1 + 5.87i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.68 - 1.55i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.14 - 7.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 6.71T + 73T^{2} \) |
| 79 | \( 1 + (-7.14 - 4.12i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.85 + 1.65i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.54iT - 89T^{2} \) |
| 97 | \( 1 + (-2.33 + 4.04i)T + (-48.5 - 84.0i)T^{2} \) |
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
−11.46871218654621440065983382851, −11.02306418877540630168917193065, −9.512222107514354476742488706306, −8.251897195447101674203981549023, −7.28301471031896091291017475235, −6.50683075187130303838320218198, −5.40151400129521856500682409938, −4.52129252903680443398895770227, −2.57559789817741945805623817585, −1.94013350551967023242217028810,
2.11243014452616380348824720038, 3.79134692270657233620590705961, 4.61944159451024953843225773959, 5.29190990039618089927038335594, 6.49075946987698039512225654832, 7.912150344520020870555411008745, 8.519915174630615732763403031473, 10.16210514827494768494486336797, 10.65029231128853891822179015375, 11.50376407613956226554107104461