| L(s) = 1 | + (−1.39 + 0.228i)2-s + (−3.32 + 0.291i)3-s + (1.89 − 0.637i)4-s + (−2.93 + 1.06i)5-s + (4.57 − 1.16i)6-s + (0.0360 − 0.0131i)7-s + (−2.50 + 1.32i)8-s + (8.04 − 1.41i)9-s + (3.85 − 2.16i)10-s + (0.601 − 2.24i)11-s + (−6.12 + 2.67i)12-s + (1.48 + 0.261i)13-s + (−0.0473 + 0.0265i)14-s + (9.45 − 4.40i)15-s + (3.18 − 2.41i)16-s + (3.57 + 2.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.986 + 0.161i)2-s + (−1.92 + 0.168i)3-s + (0.947 − 0.318i)4-s + (−1.31 + 0.477i)5-s + (1.86 − 0.476i)6-s + (0.0136 − 0.00496i)7-s + (−0.884 + 0.467i)8-s + (2.68 − 0.472i)9-s + (1.21 − 0.683i)10-s + (0.181 − 0.676i)11-s + (−1.76 + 0.771i)12-s + (0.411 + 0.0725i)13-s + (−0.0126 + 0.00709i)14-s + (2.44 − 1.13i)15-s + (0.796 − 0.604i)16-s + (0.866 + 0.607i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0427203 + 0.152382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0427203 + 0.152382i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.228i)T \) |
| 37 | \( 1 + (-5.42 - 2.74i)T \) |
| good | 3 | \( 1 + (3.32 - 0.291i)T + (2.95 - 0.520i)T^{2} \) |
| 5 | \( 1 + (2.93 - 1.06i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.0360 + 0.0131i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.601 + 2.24i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.48 - 0.261i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.57 - 2.50i)T + (5.81 + 15.9i)T^{2} \) |
| 19 | \( 1 + (0.413 + 0.346i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (0.182 + 0.680i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (2.81 + 4.87i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.84 + 2.84i)T - 31iT^{2} \) |
| 41 | \( 1 + (1.84 - 10.4i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 4.20iT - 43T^{2} \) |
| 47 | \( 1 + (10.1 + 5.85i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.17 - 6.80i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (3.42 - 9.39i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.05 + 11.6i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.20 + 6.86i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (8.34 + 7.00i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + (-5.62 - 2.62i)T + (50.7 + 60.5i)T^{2} \) |
| 83 | \( 1 + (4.03 - 5.76i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-8.87 + 4.14i)T + (57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (-0.168 - 0.627i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.17405767462407593467262215848, −10.37711930630611353609025360881, −9.598620589976331184269700893229, −8.120795133479611182863455418230, −7.54393074020113997019117174368, −6.37047413251549457880176282211, −6.05777434647081039964588017075, −4.66940230535981399024250381145, −3.47913482084072168326070543392, −1.09816538136633396578374899471,
0.20868653729893712938547629149, 1.40500209609872089121597516074, 3.72241559262756627061954654757, 4.84385160752635523956433122596, 5.85060157479618096393529574539, 7.01958460310710418932018679347, 7.42047544658525788505943224430, 8.484628004142356028610636433992, 9.726296934633335482729515759507, 10.45691253332656660842456297493
