| L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (1.61 − 3.53i)5-s + (−3.99 − 1.17i)7-s + (−0.841 + 0.540i)8-s + (3.72 − 1.09i)10-s + (2.96 − 3.42i)11-s + (3.13 − 0.919i)13-s + (−1.72 − 3.78i)14-s + (−0.959 − 0.281i)16-s + (−0.260 − 1.81i)17-s + (−0.194 + 1.35i)19-s + (3.26 + 2.10i)20-s + 4.52·22-s + (2.28 + 4.21i)23-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.0711 + 0.494i)4-s + (0.721 − 1.58i)5-s + (−1.50 − 0.443i)7-s + (−0.297 + 0.191i)8-s + (1.17 − 0.346i)10-s + (0.894 − 1.03i)11-s + (0.868 − 0.254i)13-s + (−0.462 − 1.01i)14-s + (−0.239 − 0.0704i)16-s + (−0.0631 − 0.438i)17-s + (−0.0445 + 0.310i)19-s + (0.731 + 0.469i)20-s + 0.965·22-s + (0.475 + 0.879i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61367 - 0.530057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61367 - 0.530057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-2.28 - 4.21i)T \) |
| good | 5 | \( 1 + (-1.61 + 3.53i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (3.99 + 1.17i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.96 + 3.42i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.13 + 0.919i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.260 + 1.81i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.194 - 1.35i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.0798 - 0.555i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.90 - 2.50i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.04 - 2.29i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.54 + 5.56i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-8.28 - 5.32i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 + (-10.1 - 2.98i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (6.86 - 2.01i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (2.73 - 1.76i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.11 - 4.75i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (4.22 + 4.87i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.75 - 12.2i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (8.09 - 2.37i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.156 - 0.342i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (12.7 + 8.22i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.84 + 17.1i)T + (-63.5 - 73.3i)T^{2} \) |
| 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
−11.27913988938226176262558318568, −9.942188707101335570383963744675, −9.080258932136182183560621644501, −8.657520167502856846148204131781, −7.22407760612798769342427959611, −6.05661971698254447702539865087, −5.66104131507532385973938900817, −4.24175354224245141625658363661, −3.27918015278216253357240241208, −1.02364203587348378176473845720,
2.11936519897292605700550337341, 3.09513770349596693308740979805, 4.06398097009918894390636376329, 5.88518710258358337419364852286, 6.45778145184933740559062362271, 7.10463709599284064507407193441, 9.070984380973901044327443173749, 9.660494878912159033474213612857, 10.47191723367228931624882485217, 11.18057412348778464795641020350
