| L(s) = 1 | + (−0.642 + 0.766i)2-s + (0.0781 − 0.893i)3-s + (−0.173 − 0.984i)4-s + (−2.23 − 0.164i)5-s + (0.633 + 0.633i)6-s + (−2.49 + 1.16i)7-s + (0.866 + 0.500i)8-s + (2.16 + 0.381i)9-s + (1.55 − 1.60i)10-s + (3.77 + 2.17i)11-s + (−0.893 + 0.0781i)12-s + (−1.21 + 0.213i)13-s + (0.713 − 2.66i)14-s + (−0.320 + 1.97i)15-s + (−0.939 + 0.342i)16-s + (−1.14 + 6.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.0451 − 0.515i)3-s + (−0.0868 − 0.492i)4-s + (−0.997 − 0.0733i)5-s + (0.258 + 0.258i)6-s + (−0.944 + 0.440i)7-s + (0.306 + 0.176i)8-s + (0.720 + 0.127i)9-s + (0.493 − 0.506i)10-s + (1.13 + 0.657i)11-s + (−0.257 + 0.0225i)12-s + (−0.336 + 0.0592i)13-s + (0.190 − 0.711i)14-s + (−0.0828 + 0.510i)15-s + (−0.234 + 0.0855i)16-s + (−0.277 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0142 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0142 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.537740 + 0.530116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.537740 + 0.530116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 5 | \( 1 + (2.23 + 0.164i)T \) |
| 37 | \( 1 + (1.47 - 5.90i)T \) |
| good | 3 | \( 1 + (-0.0781 + 0.893i)T + (-2.95 - 0.520i)T^{2} \) |
| 7 | \( 1 + (2.49 - 1.16i)T + (4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (-3.77 - 2.17i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.21 - 0.213i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.14 - 6.49i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.510 - 5.83i)T + (-18.7 - 3.29i)T^{2} \) |
| 23 | \( 1 + (-5.04 + 2.90i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.28 + 0.612i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (2.21 - 2.21i)T - 31iT^{2} \) |
| 41 | \( 1 + (6.65 - 1.17i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 5.81iT - 43T^{2} \) |
| 47 | \( 1 + (7.97 + 2.13i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.992 - 0.462i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-4.09 + 8.77i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-6.70 - 4.69i)T + (20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (5.89 + 12.6i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-6.23 + 5.23i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (5.63 - 5.63i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.99 - 2.79i)T + (50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (-4.85 + 3.39i)T + (28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-3.50 - 1.63i)T + (57.2 + 68.1i)T^{2} \) |
| 97 | \( 1 + (-4.73 - 8.20i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.82850601865765854544589637704, −10.49564700260849136637759702055, −9.709536206860912480934780883828, −8.649607773005380697141599892427, −7.902498959036812673945321025561, −6.76294228021971444843847321779, −6.42512623521756295052587102020, −4.68933177994762730163490439272, −3.58124267459380363294892523172, −1.57427147634513774716540762654,
0.64020564153646379320047111055, 3.05250792080593091378680843771, 3.83806725991808490140260581620, 4.86931721562029037189666489938, 6.96301459883658772530843875410, 7.13158158698347284125922131242, 8.815357849048776463081508993150, 9.312224602140195215771836449369, 10.22052601400565741193396395539, 11.25321852050723645082311841776
