| L(s) = 1 | + (−0.254 − 0.493i)3-s + (1.20 − 0.114i)5-s + (1.86 − 1.88i)7-s + (1.56 − 2.19i)9-s + (0.0516 − 0.0178i)11-s + (1.70 + 5.79i)13-s + (−0.362 − 0.563i)15-s + (−6.74 − 2.70i)17-s + (2.94 − 1.18i)19-s + (−1.40 − 0.439i)21-s + (3.30 − 3.47i)23-s + (−3.47 + 0.670i)25-s + (−3.12 − 0.449i)27-s + (−0.826 − 5.74i)29-s + (9.41 − 0.448i)31-s + ⋯ |
| L(s) = 1 | + (−0.146 − 0.284i)3-s + (0.537 − 0.0513i)5-s + (0.703 − 0.710i)7-s + (0.520 − 0.730i)9-s + (0.0155 − 0.00538i)11-s + (0.472 + 1.60i)13-s + (−0.0935 − 0.145i)15-s + (−1.63 − 0.654i)17-s + (0.676 − 0.270i)19-s + (−0.305 − 0.0958i)21-s + (0.689 − 0.724i)23-s + (−0.695 + 0.134i)25-s + (−0.601 − 0.0865i)27-s + (−0.153 − 1.06i)29-s + (1.69 − 0.0805i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55477 - 0.718012i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55477 - 0.718012i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.86 + 1.88i)T \) |
| 23 | \( 1 + (-3.30 + 3.47i)T \) |
| good | 3 | \( 1 + (0.254 + 0.493i)T + (-1.74 + 2.44i)T^{2} \) |
| 5 | \( 1 + (-1.20 + 0.114i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-0.0516 + 0.0178i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 5.79i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (6.74 + 2.70i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-2.94 + 1.18i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.826 + 5.74i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-9.41 + 0.448i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (0.587 + 0.418i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-6.80 - 3.10i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (2.60 - 4.05i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-4.17 + 2.40i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.25 - 6.55i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-1.97 - 0.479i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (9.06 + 4.67i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (0.0293 + 0.152i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (6.77 + 7.82i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (3.21 - 4.08i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (5.33 - 5.59i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.34 + 2.95i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.140 + 2.95i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (2.28 - 5.00i)T + (-63.5 - 73.3i)T^{2} \) |
| 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
−10.46907117377977629450038273943, −9.444623305624949252014774143404, −8.928638670556697200665751407865, −7.65988508708883393101525729488, −6.76947444600238451567563582279, −6.22856382814186175636279469670, −4.67389861948515143084795550075, −4.12849367894551772878379574869, −2.33831126347265660380847984704, −1.09469998602010038262889290425,
1.61029845843019899909304607562, 2.82125207136868867749381526383, 4.30510652140900794821085108121, 5.32598345108199181275132052813, 5.89769121547150738211475837870, 7.22553585260364018067069874473, 8.180674412366787980263429031251, 8.882130086210351817666366646643, 9.980546431600431316570894082530, 10.66927200503125204375136623203
