L(s) = 1 | + (0.609 + 1.05i)2-s + (−1.16 − 2.01i)3-s + (0.256 − 0.443i)4-s + 5-s + (1.42 − 2.46i)6-s + (1.80 − 3.11i)7-s + 3.06·8-s + (−1.21 + 2.11i)9-s + (0.609 + 1.05i)10-s + (2.68 + 4.65i)11-s − 1.19·12-s + 4.39·14-s + (−1.16 − 2.01i)15-s + (1.35 + 2.34i)16-s + (0.565 − 0.980i)17-s − 2.97·18-s + ⋯ |
L(s) = 1 | + (0.431 + 0.746i)2-s + (−0.673 − 1.16i)3-s + (0.128 − 0.221i)4-s + 0.447·5-s + (0.580 − 1.00i)6-s + (0.680 − 1.17i)7-s + 1.08·8-s + (−0.406 + 0.704i)9-s + (0.192 + 0.334i)10-s + (0.809 + 1.40i)11-s − 0.344·12-s + 1.17·14-s + (−0.301 − 0.521i)15-s + (0.339 + 0.587i)16-s + (0.137 − 0.237i)17-s − 0.701·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96100 - 0.879720i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96100 - 0.879720i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.609 - 1.05i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.16 + 2.01i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.80 + 3.11i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.68 - 4.65i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.565 + 0.980i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.13 + 1.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.94 - 3.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0123 - 0.0214i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + (4.35 + 7.53i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.86 - 3.23i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.565 + 0.980i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + (0.0857 - 0.148i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.68 - 2.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.19 - 5.54i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.39 - 9.35i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + (8.07 + 13.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.08 + 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.18507902678104212171431632084, −9.287037065071710729683137167027, −7.72822937599521302445570604163, −7.14064877028083538329558708151, −6.88708197171669483305212771211, −5.81976953194234687206472772494, −5.00272765370528268542925052338, −4.08934130050012725805770056991, −1.87377430814152765989430284289, −1.17730105056341723671484210050,
1.67002722809890081136681214572, 3.01123355536095690793032785882, 3.88537009694859947603213335699, 4.94187778006346245328341778709, 5.57717250914804363721917138575, 6.48748829030025526129875746401, 8.030999114057250603967206853924, 8.824747787006173835724895781461, 9.632546257013516428733655399346, 10.73122192386195676827205473871