| L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (1.18 + 0.348i)5-s + (0.415 − 0.909i)6-s + (0.968 − 1.11i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.809 + 0.934i)10-s + (−0.745 + 0.479i)11-s + (0.841 − 0.540i)12-s + (−0.0440 − 0.0508i)13-s + (1.41 − 0.416i)14-s + (0.175 − 1.22i)15-s + (−0.654 + 0.755i)16-s + (−1.78 + 3.90i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (0.530 + 0.155i)5-s + (0.169 − 0.371i)6-s + (0.366 − 0.422i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.256 + 0.295i)10-s + (−0.224 + 0.144i)11-s + (0.242 − 0.156i)12-s + (−0.0122 − 0.0141i)13-s + (0.379 − 0.111i)14-s + (0.0454 − 0.316i)15-s + (−0.163 + 0.188i)16-s + (−0.432 + 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50801 + 0.112173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50801 + 0.112173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.72 + 0.847i)T \) |
| good | 5 | \( 1 + (-1.18 - 0.348i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.968 + 1.11i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (0.745 - 0.479i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.0440 + 0.0508i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.78 - 3.90i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.93 + 4.24i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.18 - 6.98i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.226 + 1.57i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.01 + 1.47i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.130 - 0.0382i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.936 + 6.51i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 8.31T + 47T^{2} \) |
| 53 | \( 1 + (-1.96 + 2.26i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.47 - 5.16i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.654 + 4.55i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 6.70i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (6.15 + 3.95i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.48 - 12.0i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (5.16 + 5.96i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-8.06 + 2.36i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.66 + 11.6i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-16.4 - 4.83i)T + (81.6 + 52.4i)T^{2} \) |
| 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
−13.28109907142700945584162785112, −12.50720236791205523149987753630, −11.30070720967670445350579604234, −10.33981863521739536853323294859, −8.791157565635583691698588518295, −7.65419741008514171148048620590, −6.62042749934659240512059713428, −5.58801692398529840422954548835, −4.14405422002065102148624426020, −2.21667700905905247225843206163,
2.27902221926564689561934401534, 3.99709817693010872171911577506, 5.26717863639586492164341322026, 6.16254776015428022672530349365, 7.946401957481759560254219009577, 9.320505957667692781607219845477, 10.11885445574700742608266810200, 11.27847933147793583056103122633, 12.04229824782263472581508108515, 13.26048784852090024459164075442
