L(s) = 1 | + (0.766 − 0.642i)2-s + (−1.90 − 1.59i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s − 2.48·6-s + (−4.17 + 1.51i)7-s + (−0.500 − 0.866i)8-s + (0.551 + 3.12i)9-s + (−0.5 + 0.866i)10-s + (2.64 + 4.57i)11-s + (−1.90 + 1.59i)12-s + (−0.291 + 1.65i)13-s + (−2.22 + 3.84i)14-s + (2.33 + 0.849i)15-s + (−0.939 − 0.342i)16-s + (−1.09 − 6.22i)17-s + ⋯ |
L(s) = 1 | + (0.541 − 0.454i)2-s + (−1.09 − 0.922i)3-s + (0.0868 − 0.492i)4-s + (−0.420 + 0.152i)5-s − 1.01·6-s + (−1.57 + 0.574i)7-s + (−0.176 − 0.306i)8-s + (0.183 + 1.04i)9-s + (−0.158 + 0.273i)10-s + (0.797 + 1.38i)11-s + (−0.549 + 0.461i)12-s + (−0.0809 + 0.458i)13-s + (−0.593 + 1.02i)14-s + (0.602 + 0.219i)15-s + (−0.234 − 0.0855i)16-s + (−0.266 − 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0580168 + 0.0733564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0580168 + 0.0733564i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.939 - 0.342i)T \) |
| 37 | \( 1 + (5.41 - 2.77i)T \) |
good | 3 | \( 1 + (1.90 + 1.59i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (4.17 - 1.51i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.64 - 4.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.291 - 1.65i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.09 + 6.22i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.698 + 0.586i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (2.72 - 4.72i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.39 + 5.88i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.00T + 31T^{2} \) |
| 41 | \( 1 + (1.27 - 7.24i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + 2.18T + 43T^{2} \) |
| 47 | \( 1 + (1.72 - 2.98i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.95 + 0.710i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (8.25 + 3.00i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.910 + 5.16i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.61 + 1.68i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.18 - 5.19i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 8.95T + 73T^{2} \) |
| 79 | \( 1 + (-3.47 + 1.26i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.03 + 17.2i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (5.71 + 2.07i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-4.75 + 8.24i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.78247056166479532778716029055, −11.29187647002460385411719814959, −9.751414463425368220491979309407, −9.376945073968584961021239209778, −7.34211049885596498108597431450, −6.76546719843733233714711216385, −6.01647630044507659979845561228, −4.84955061058683504908846016059, −3.48334692065827791487743711091, −1.97178130974407670438710996757,
0.05756900099449319465007910553, 3.64324452276355741562977550812, 3.85989973936150435417898234323, 5.41862303523444359082887577965, 6.15288096031129119116966269507, 6.87547486373305979872538826945, 8.392905575814330320325647002298, 9.360977018231391520093220715775, 10.65262751077131653616002767203, 10.83697976064933602224942737863