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
−11.25321852050723645082311841776, −10.22052601400565741193396395539, −9.312224602140195215771836449369, −8.815357849048776463081508993150, −7.13158158698347284125922131242, −6.96301459883658772530843875410, −4.86931721562029037189666489938, −3.83806725991808490140260581620, −3.05250792080593091378680843771, −0.64020564153646379320047111055,
1.57427147634513774716540762654, 3.58124267459380363294892523172, 4.68933177994762730163490439272, 6.42512623521756295052587102020, 6.76294228021971444843847321779, 7.902498959036812673945321025561, 8.649607773005380697141599892427, 9.709536206860912480934780883828, 10.49564700260849136637759702055, 11.82850601865765854544589637704