L(s) = 1 | + (0.565 − 0.979i)2-s + (0.360 + 0.623i)4-s + (−0.634 + 1.09i)5-s + (2.62 − 0.337i)7-s + 3.07·8-s + (0.718 + 1.24i)10-s + (−1.90 − 3.29i)11-s + 2.00·13-s + (1.15 − 2.76i)14-s + (1.01 − 1.76i)16-s + (0.5 + 0.866i)17-s + (−0.436 + 0.756i)19-s − 0.915·20-s − 4.30·22-s + (−0.799 + 1.38i)23-s + ⋯ |
L(s) = 1 | + (0.399 − 0.692i)2-s + (0.180 + 0.311i)4-s + (−0.283 + 0.491i)5-s + (0.991 − 0.127i)7-s + 1.08·8-s + (0.227 + 0.393i)10-s + (−0.573 − 0.994i)11-s + 0.555·13-s + (0.308 − 0.738i)14-s + (0.254 − 0.441i)16-s + (0.121 + 0.210i)17-s + (−0.100 + 0.173i)19-s − 0.204·20-s − 0.918·22-s + (−0.166 + 0.288i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.442335026\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.442335026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.337i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.565 + 0.979i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (0.634 - 1.09i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.90 + 3.29i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 19 | \( 1 + (0.436 - 0.756i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.799 - 1.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + (2.88 + 4.98i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.41 - 4.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.84T + 41T^{2} \) |
| 43 | \( 1 - 6.29T + 43T^{2} \) |
| 47 | \( 1 + (0.128 - 0.223i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.44 - 4.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.17 - 2.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.510 - 0.884i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.46 + 9.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.690T + 71T^{2} \) |
| 73 | \( 1 + (2.39 + 4.15i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.43 + 7.68i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + (2.73 - 4.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−10.41969060941402560544646802608, −8.872892172693681944915168174844, −8.074825909671282781045065401896, −7.56462457872090310953555830293, −6.47464443330468207864718243116, −5.37054882870995995344312198149, −4.37032257315192574506895093084, −3.46598418871261744182046994795, −2.63494067353032041442670671575, −1.33330759586967808574176487578,
1.22848695214762540759411913147, 2.42473668722350834689711964872, 4.21568597696625655612721574558, 4.85235161942242979939272074902, 5.49041699110360442302019982418, 6.60883505958085990062881053368, 7.34508047694269035921823604802, 8.183233245406727785848512519654, 8.812382833118755645399840822242, 10.14657273359372244930360386728