| L(s) = 1 | + (0.249 + 1.39i)2-s + (−0.471 − 0.881i)3-s + (−1.87 + 0.693i)4-s + (0.986 − 0.809i)5-s + (1.11 − 0.875i)6-s + (1.86 − 1.24i)7-s + (−1.43 − 2.43i)8-s + (−0.555 + 0.831i)9-s + (1.37 + 1.17i)10-s + (−0.801 − 2.64i)11-s + (1.49 + 1.32i)12-s + (0.812 − 0.989i)13-s + (2.20 + 2.29i)14-s + (−1.17 − 0.488i)15-s + (3.03 − 2.60i)16-s + (6.22 − 2.57i)17-s + ⋯ |
| L(s) = 1 | + (0.176 + 0.984i)2-s + (−0.272 − 0.509i)3-s + (−0.937 + 0.346i)4-s + (0.441 − 0.361i)5-s + (0.453 − 0.357i)6-s + (0.706 − 0.472i)7-s + (−0.506 − 0.862i)8-s + (−0.185 + 0.277i)9-s + (0.434 + 0.370i)10-s + (−0.241 − 0.796i)11-s + (0.431 + 0.383i)12-s + (0.225 − 0.274i)13-s + (0.589 + 0.612i)14-s + (−0.304 − 0.126i)15-s + (0.759 − 0.650i)16-s + (1.50 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38112 + 0.00819568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38112 + 0.00819568i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.249 - 1.39i)T \) |
| 3 | \( 1 + (0.471 + 0.881i)T \) |
| good | 5 | \( 1 + (-0.986 + 0.809i)T + (0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.86 + 1.24i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.801 + 2.64i)T + (-9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-0.812 + 0.989i)T + (-2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-6.22 + 2.57i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (2.45 - 0.241i)T + (18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-4.61 + 0.917i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-3.05 - 0.926i)T + (24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (1.35 + 1.35i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.392 - 3.98i)T + (-36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.0227 - 0.114i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.604 + 1.13i)T + (-23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (3.37 + 8.15i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (7.45 - 2.26i)T + (44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (6.40 + 7.79i)T + (-11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (0.976 - 0.521i)T + (33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-0.00283 + 0.00151i)T + (37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (0.720 + 1.07i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.09 - 3.40i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (5.69 - 13.7i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.758 + 7.70i)T + (-81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-16.8 - 3.34i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (1.56 + 1.56i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.42442067713761291949749140850, −10.38496411065280567273193318975, −9.258303787839630950295269640907, −8.250483233024770200708760520002, −7.63139500694559145120789881641, −6.55783197385309282218352649398, −5.53462659350277732287892698339, −4.88670931626040109439361260757, −3.31500627193320149016148681717, −1.05203799271135417318229091075,
1.69577825148778245484343440828, 3.01487011461153053335841023467, 4.35935608130073378480481225502, 5.24611764557826380464906700119, 6.21170659868905327339899903850, 7.85123023315079950567662627836, 8.891103030362553449137163141208, 9.811668291710731086178619716493, 10.48918963329775355453965845708, 11.22525958854334950248309449805
