| L(s) = 1 | + (−0.674 + 0.502i)2-s + (−0.868 − 1.49i)3-s + (−0.370 + 1.23i)4-s + (0.155 + 2.67i)5-s + (1.33 + 0.574i)6-s + (−0.412 − 2.61i)7-s + (−0.946 − 2.60i)8-s + (−1.49 + 2.60i)9-s + (−1.44 − 1.72i)10-s + (3.41 − 5.19i)11-s + (2.17 − 0.519i)12-s + (0.140 + 0.104i)13-s + (1.59 + 1.55i)14-s + (3.86 − 2.55i)15-s + (−0.216 − 0.142i)16-s + (0.929 + 5.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.476 + 0.354i)2-s + (−0.501 − 0.865i)3-s + (−0.185 + 0.619i)4-s + (0.0695 + 1.19i)5-s + (0.546 + 0.234i)6-s + (−0.155 − 0.987i)7-s + (−0.334 − 0.919i)8-s + (−0.497 + 0.867i)9-s + (−0.457 − 0.544i)10-s + (1.02 − 1.56i)11-s + (0.628 − 0.149i)12-s + (0.0390 + 0.0290i)13-s + (0.424 + 0.415i)14-s + (0.998 − 0.658i)15-s + (−0.0540 − 0.0355i)16-s + (0.225 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.841223 + 0.295990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841223 + 0.295990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.868 + 1.49i)T \) |
| 7 | \( 1 + (0.412 + 2.61i)T \) |
| good | 2 | \( 1 + (0.674 - 0.502i)T + (0.573 - 1.91i)T^{2} \) |
| 5 | \( 1 + (-0.155 - 2.67i)T + (-4.96 + 0.580i)T^{2} \) |
| 11 | \( 1 + (-3.41 + 5.19i)T + (-4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.140 - 0.104i)T + (3.72 + 12.4i)T^{2} \) |
| 17 | \( 1 + (-0.929 - 5.26i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-2.08 - 0.367i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.91 - 8.08i)T + (-20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-1.97 - 0.853i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (0.362 + 1.52i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (5.15 - 4.32i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-10.0 - 1.17i)T + (39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (-0.704 - 0.463i)T + (17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (-7.25 - 1.71i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + 2.72iT - 53T^{2} \) |
| 59 | \( 1 + (-6.97 - 3.50i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-11.8 + 3.55i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (10.7 + 1.26i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-1.77 + 4.86i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.48 - 0.261i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.692 + 0.930i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (7.88 - 0.921i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (-6.85 - 5.75i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.931 - 1.85i)T + (-57.9 + 77.8i)T^{2} \) |
| 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
−10.99928884812730255046245433786, −10.04497793738259229140589052015, −8.862022599997922353446245943195, −7.935120926082430476921092691966, −7.22891606787233979137386384381, −6.55361213737877291732596383795, −5.83047302862388399878885697092, −3.84520681525575441058227220886, −3.14654655815631884081984300677, −1.08031015946105689758679811131,
0.853819251599963385385573193718, 2.43158476225745156519958601776, 4.36551696564719438047961157649, 4.99366265264393711251729317191, 5.73101581630373501677851136105, 6.91647387577983084929148449172, 8.679352460628846663824112661581, 9.115196339490506091728622525628, 9.622167203960185912040193242264, 10.41508520910099324445570818733
