| L(s) = 1 | + (−0.664 − 1.24i)2-s + (2.34 − 1.25i)3-s + (−1.11 + 1.65i)4-s + (−2.35 − 2.86i)5-s + (−3.12 − 2.09i)6-s + (−1.77 + 1.18i)7-s + (2.81 + 0.293i)8-s + (2.27 − 3.40i)9-s + (−2.01 + 4.83i)10-s + (3.00 − 0.911i)11-s + (−0.542 + 5.29i)12-s + (4.40 + 3.61i)13-s + (2.66 + 1.42i)14-s + (−9.11 − 3.77i)15-s + (−1.50 − 3.70i)16-s + (−2.12 + 0.878i)17-s + ⋯ |
| L(s) = 1 | + (−0.469 − 0.882i)2-s + (1.35 − 0.724i)3-s + (−0.558 + 0.829i)4-s + (−1.05 − 1.28i)5-s + (−1.27 − 0.856i)6-s + (−0.670 + 0.448i)7-s + (0.994 + 0.103i)8-s + (0.757 − 1.13i)9-s + (−0.636 + 1.52i)10-s + (0.905 − 0.274i)11-s + (−0.156 + 1.52i)12-s + (1.22 + 1.00i)13-s + (0.710 + 0.381i)14-s + (−2.35 − 0.975i)15-s + (−0.375 − 0.926i)16-s + (−0.514 + 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.554993 - 0.889329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.554993 - 0.889329i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.664 + 1.24i)T \) |
| good | 3 | \( 1 + (-2.34 + 1.25i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (2.35 + 2.86i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (1.77 - 1.18i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-3.00 + 0.911i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-4.40 - 3.61i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (2.12 - 0.878i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.222 + 2.26i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-4.14 + 0.825i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.05 - 3.49i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (0.733 + 0.733i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.40 - 0.138i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.67 - 8.43i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (2.27 + 1.21i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (4.22 + 10.1i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.810 + 2.67i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (6.48 - 5.32i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-4.18 - 7.83i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-1.02 - 1.91i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (4.85 + 7.27i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (9.35 + 6.24i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (1.50 - 3.63i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.375 - 0.0370i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.47 - 1.09i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (4.20 + 4.20i)T + 97iT^{2} \) |
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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
−13.07079803679315755105087019683, −12.08290901244520397635163118864, −11.27068879638800135154018140752, −9.172917409422196156005829111429, −8.935571015720606109831312036886, −8.209409398388308390904653841545, −6.87805536194873879411404066097, −4.29036708705534102643096987962, −3.26803491380099847273560936996, −1.42406646491903993672052035116,
3.29517762567698438732748443267, 4.09970289458848895449310038703, 6.38041621831302677802933936720, 7.40694169568954725407276968411, 8.289133662027932321021558408505, 9.319655658887929925596786244938, 10.30347859547921159627852664887, 11.13729187316516548631608946029, 13.15921021234804802666041766489, 14.18695124098189422693655743428
