| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.400 + 1.10i)3-s + (0.766 + 0.642i)4-s + (−0.822 + 2.07i)5-s − 1.17i·6-s + (3.81 + 0.672i)7-s + (−0.500 − 0.866i)8-s + (1.24 − 1.04i)9-s + (1.48 − 1.67i)10-s + (−1.57 − 2.73i)11-s + (−0.400 + 1.10i)12-s + (1.40 + 1.17i)13-s + (−3.35 − 1.93i)14-s + (−2.61 − 0.0716i)15-s + (0.173 + 0.984i)16-s + (−4.46 + 3.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.231 + 0.635i)3-s + (0.383 + 0.321i)4-s + (−0.367 + 0.929i)5-s − 0.477i·6-s + (1.44 + 0.254i)7-s + (−0.176 − 0.306i)8-s + (0.416 − 0.349i)9-s + (0.469 − 0.529i)10-s + (−0.476 − 0.825i)11-s + (−0.115 + 0.317i)12-s + (0.389 + 0.326i)13-s + (−0.896 − 0.517i)14-s + (−0.675 − 0.0185i)15-s + (0.0434 + 0.246i)16-s + (−1.08 + 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.966980 + 0.630429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.966980 + 0.630429i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (0.822 - 2.07i)T \) |
| 37 | \( 1 + (5.80 - 1.80i)T \) |
| good | 3 | \( 1 + (-0.400 - 1.10i)T + (-2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-3.81 - 0.672i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.57 + 2.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.40 - 1.17i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.46 - 3.74i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-2.67 - 7.36i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (0.0262 - 0.0453i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.42 + 1.97i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.88iT - 31T^{2} \) |
| 41 | \( 1 + (-2.75 - 2.30i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + 2.80T + 43T^{2} \) |
| 47 | \( 1 + (9.42 + 5.44i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.8 + 2.25i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-12.1 + 2.13i)T + (55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.0728 + 0.0867i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.55 + 0.803i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (12.4 - 4.52i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + 4.68iT - 73T^{2} \) |
| 79 | \( 1 + (5.72 + 1.01i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.100 + 0.119i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.80 + 0.493i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.58 + 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.38683522007140539924936791557, −10.54679465509251843419695932294, −10.00073776361092620657816907421, −8.536963662261877030814553770238, −8.240298203689834128493820960975, −7.03611333186100911068321055422, −5.85444566358415927966452744958, −4.31076573604547779823843770199, −3.36283679114922850664636325709, −1.81013326096487212169949655896,
1.06589462739155474438159066721, 2.28613505119670152909331923335, 4.63459998677372287533252018930, 5.09184022539829798399677701454, 7.03022162098182473912832533017, 7.48795124167143454396000540605, 8.412548255977344607215051981176, 9.029037217990942661628778415248, 10.32142202456223859438620550750, 11.25405299159133976774297558727
