| L(s) = 1 | + (−1.40 + 0.135i)2-s + (−0.342 + 0.939i)3-s + (1.96 − 0.381i)4-s + (2.71 − 0.479i)5-s + (0.353 − 1.36i)6-s + (−1.39 + 2.41i)7-s + (−2.71 + 0.803i)8-s + (−0.766 − 0.642i)9-s + (−3.76 + 1.04i)10-s + (−5.27 + 3.04i)11-s + (−0.312 + 1.97i)12-s + (0.786 + 2.16i)13-s + (1.63 − 3.59i)14-s + (−0.479 + 2.71i)15-s + (3.70 − 1.49i)16-s + (2.16 − 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 + 0.0959i)2-s + (−0.197 + 0.542i)3-s + (0.981 − 0.190i)4-s + (1.21 − 0.214i)5-s + (0.144 − 0.558i)6-s + (−0.527 + 0.913i)7-s + (−0.958 + 0.284i)8-s + (−0.255 − 0.214i)9-s + (−1.18 + 0.330i)10-s + (−1.58 + 0.917i)11-s + (−0.0902 + 0.570i)12-s + (0.218 + 0.599i)13-s + (0.437 − 0.959i)14-s + (−0.123 + 0.702i)15-s + (0.927 − 0.374i)16-s + (0.524 − 0.440i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.372213 + 0.649652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.372213 + 0.649652i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.135i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (-0.359 - 4.34i)T \) |
| good | 5 | \( 1 + (-2.71 + 0.479i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.39 - 2.41i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.27 - 3.04i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.786 - 2.16i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.16 + 1.81i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.07 + 6.12i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (4.49 - 5.36i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.36 - 4.09i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.19iT - 37T^{2} \) |
| 41 | \( 1 + (-0.915 - 0.333i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-9.76 + 1.72i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.90 + 1.60i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.92 + 0.515i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (4.93 + 5.88i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (5.77 + 1.01i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.91 + 2.27i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.854 - 4.84i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.15 - 2.24i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.75 - 0.638i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 7.09i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.58 - 0.940i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (1.53 - 1.28i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.93174366300566115074113259585, −10.22310043585931390753386947032, −9.595023952702292405753545527812, −8.991118111266376532651283542538, −7.923293791493214019045401806233, −6.69308905110682518230625724606, −5.75056947905790533601914046049, −5.08308791849330387320237009185, −2.93405134455026192843981650926, −1.92409075824632816128701356124,
0.61649941673057203690136954120, 2.22828195403798296238066160307, 3.29963089693539945052746815658, 5.65576906021724915422555934214, 6.03032047373643493850558187137, 7.39584033819323408372779857284, 7.80673717643490581524238505196, 9.150415339412068383119755977352, 9.911831081782419612435186246179, 10.75191224487137922083306410092
