| L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.971 + 0.432i)5-s + (0.809 + 0.587i)6-s + (−2.64 + 0.0543i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.531 + 0.920i)10-s + (3.31 − 0.181i)11-s + (0.5 − 0.866i)12-s + (4.60 − 3.34i)13-s + (0.330 + 2.62i)14-s + (0.328 − 1.01i)15-s + (0.913 − 0.406i)16-s + (0.389 − 3.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (−0.386 + 0.429i)3-s + (−0.489 + 0.103i)4-s + (−0.434 + 0.193i)5-s + (0.330 + 0.239i)6-s + (−0.999 + 0.0205i)7-s + (0.109 + 0.336i)8-s + (−0.0348 − 0.331i)9-s + (0.168 + 0.291i)10-s + (0.998 − 0.0547i)11-s + (0.144 − 0.249i)12-s + (1.27 − 0.927i)13-s + (0.0883 + 0.701i)14-s + (0.0848 − 0.261i)15-s + (0.228 − 0.101i)16-s + (0.0945 − 0.899i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.939990 - 0.412752i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939990 - 0.412752i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (2.64 - 0.0543i)T \) |
| 11 | \( 1 + (-3.31 + 0.181i)T \) |
| good | 5 | \( 1 + (0.971 - 0.432i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-4.60 + 3.34i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.389 + 3.70i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-8.21 - 1.74i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.15 + 2.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.545 - 1.68i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.70 - 2.98i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.81 - 2.01i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (3.16 + 9.73i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 + (1.46 + 0.311i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (7.86 + 3.50i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-10.5 + 2.23i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.592 + 0.263i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (3.72 + 6.45i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.44 + 6.86i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.56 + 0.758i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-0.786 - 7.48i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-8.70 - 6.32i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (7.26 - 12.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.1 + 9.54i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.00920396529785097196512534809, −10.03085657077965301251590631182, −9.438530415552923537656929432528, −8.475535831141322060967530305714, −7.22240011899998516774004633808, −6.15092824807924973238915266980, −5.11486305047920809213397085681, −3.61771636815539009155943331014, −3.24166292167420013893295015617, −0.920197941021097557436734638545,
1.16490214832631486138491500099, 3.44340767999179467750626635610, 4.42476879602237252251148516277, 5.94273527193292316486009960986, 6.42773464320600501303367983994, 7.35549007738493011503313102328, 8.371841072532961588918365822537, 9.277265257531965764335238260184, 10.04237245916926969924484570249, 11.52338860145863739375800241627
