| L(s) = 1 | + (−0.642 + 0.766i)2-s + (1.19 + 1.25i)3-s + (−0.173 − 0.984i)4-s + (−2.23 + 0.138i)5-s + (−1.72 + 0.110i)6-s + (2.23 − 1.28i)7-s + (0.866 + 0.500i)8-s + (−0.140 + 2.99i)9-s + (1.32 − 1.79i)10-s + (2.63 + 1.51i)11-s + (1.02 − 1.39i)12-s + (3.00 + 1.09i)13-s + (−0.447 + 2.53i)14-s + (−2.84 − 2.63i)15-s + (−0.939 + 0.342i)16-s + (−1.98 − 1.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.690 + 0.723i)3-s + (−0.0868 − 0.492i)4-s + (−0.998 + 0.0620i)5-s + (−0.705 + 0.0450i)6-s + (0.843 − 0.487i)7-s + (0.306 + 0.176i)8-s + (−0.0469 + 0.998i)9-s + (0.420 − 0.568i)10-s + (0.793 + 0.458i)11-s + (0.296 − 0.402i)12-s + (0.832 + 0.302i)13-s + (−0.119 + 0.678i)14-s + (−0.733 − 0.679i)15-s + (−0.234 + 0.0855i)16-s + (−0.480 − 0.403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.792238 + 1.07320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.792238 + 1.07320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 + (-1.19 - 1.25i)T \) |
| 5 | \( 1 + (2.23 - 0.138i)T \) |
| 19 | \( 1 + (0.765 - 4.29i)T \) |
| good | 7 | \( 1 + (-2.23 + 1.28i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.63 - 1.51i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.00 - 1.09i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.98 + 1.66i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.878 - 4.98i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.575 - 0.482i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (7.47 - 4.31i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.30T + 37T^{2} \) |
| 41 | \( 1 + (0.243 - 0.0887i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (5.12 + 0.903i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.62 + 5.55i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-4.33 + 0.764i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (3.10 + 2.60i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.76 + 10.0i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.42 + 7.91i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.792 - 4.49i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.36 - 9.24i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.67 - 15.5i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.188 - 0.326i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (12.2 + 4.47i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.74 + 1.46i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.07886325912111295769753148662, −9.957379407926717615316959824160, −9.058245791075941174732373671907, −8.357561755515241327513894238428, −7.62433499656817867410406345101, −6.84186072903234921268223351342, −5.30700210679982043808445516564, −4.26302795392755809978075366908, −3.61433351426831022323671407982, −1.64185660789374539542583642992,
0.915391716790841517600557862411, 2.30473082965315378374803660380, 3.51576530187035796456282340388, 4.44376080899122861206516050594, 6.13209267303969254433928573778, 7.19484472140445165025154759109, 8.076479336201596053606772226890, 8.705063914404411637950963808007, 9.160329976070735703071072812145, 10.79993521781754730995531887978
