| L(s) = 1 | + (0.823 + 1.15i)2-s + (1.18 − 0.635i)3-s + (−0.645 + 1.89i)4-s + (−0.634 − 0.773i)5-s + (1.70 + 0.843i)6-s + (−4.07 + 2.72i)7-s + (−2.70 + 0.815i)8-s + (−0.658 + 0.985i)9-s + (0.366 − 1.36i)10-s + (−5.20 + 1.57i)11-s + (0.435 + 2.65i)12-s + (3.21 + 2.63i)13-s + (−6.48 − 2.44i)14-s + (−1.24 − 0.515i)15-s + (−3.16 − 2.44i)16-s + (3.95 − 1.64i)17-s + ⋯ |
| L(s) = 1 | + (0.581 + 0.813i)2-s + (0.686 − 0.366i)3-s + (−0.322 + 0.946i)4-s + (−0.283 − 0.345i)5-s + (0.697 + 0.344i)6-s + (−1.53 + 1.02i)7-s + (−0.957 + 0.288i)8-s + (−0.219 + 0.328i)9-s + (0.116 − 0.431i)10-s + (−1.56 + 0.476i)11-s + (0.125 + 0.767i)12-s + (0.891 + 0.731i)13-s + (−1.73 − 0.653i)14-s + (−0.321 − 0.133i)15-s + (−0.791 − 0.610i)16-s + (0.960 − 0.397i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.133329 + 1.22255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133329 + 1.22255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.823 - 1.15i)T \) |
| 5 | \( 1 + (0.634 + 0.773i)T \) |
| good | 3 | \( 1 + (-1.18 + 0.635i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (4.07 - 2.72i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (5.20 - 1.57i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-3.21 - 2.63i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-3.95 + 1.64i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.725 + 7.36i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (0.125 - 0.0249i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.673 - 2.21i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-6.08 - 6.08i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.21 + 0.217i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-2.29 - 11.5i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (1.39 + 0.745i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.64 - 6.39i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.26 - 4.17i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (1.98 - 1.62i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (0.230 + 0.431i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (4.49 + 8.40i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-3.72 - 5.57i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-3.99 - 2.67i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (2.35 - 5.69i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (6.14 - 0.605i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (2.48 + 0.494i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (1.35 + 1.35i)T + 97iT^{2} \) |
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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
−11.14858746639544451352611397790, −9.736771426986255949513466332547, −8.922340875272599660743467386861, −8.343927097388516228014576851229, −7.39832742693370488501814966539, −6.57063464432237178493167061170, −5.55895532031661480005499963756, −4.72136070917921146273347353691, −3.10061309992106238358794899013, −2.70263923327728635300213512438,
0.48101149615792726524400469515, 2.70384711750067323790100521499, 3.57380600132465621610164205031, 3.82468646294150955989788406262, 5.67447187367798370110536679174, 6.22614288097513270996430455462, 7.68478297833541464014902334665, 8.503806691494550901997262566078, 9.777040220500500792859341644845, 10.28010006349828799301153543596
