| L(s) = 1 | + (0.199 − 0.0424i)2-s + (−1.41 − 0.998i)3-s + (−1.78 + 0.796i)4-s + (2.05 + 0.891i)5-s + (−0.325 − 0.139i)6-s + (0.530 + 0.918i)7-s + (−0.654 + 0.475i)8-s + (1.00 + 2.82i)9-s + (0.447 + 0.0909i)10-s + (5.73 − 1.21i)11-s + (3.32 + 0.658i)12-s + (3.20 + 0.680i)13-s + (0.144 + 0.160i)14-s + (−2.01 − 3.30i)15-s + (2.51 − 2.78i)16-s + (−5.93 + 4.31i)17-s + ⋯ |
| L(s) = 1 | + (0.141 − 0.0300i)2-s + (−0.817 − 0.576i)3-s + (−0.894 + 0.398i)4-s + (0.917 + 0.398i)5-s + (−0.132 − 0.0568i)6-s + (0.200 + 0.346i)7-s + (−0.231 + 0.168i)8-s + (0.335 + 0.941i)9-s + (0.141 + 0.0287i)10-s + (1.72 − 0.367i)11-s + (0.960 + 0.190i)12-s + (0.887 + 0.188i)13-s + (0.0387 + 0.0430i)14-s + (−0.519 − 0.854i)15-s + (0.627 − 0.696i)16-s + (−1.43 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03545 + 0.192185i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03545 + 0.192185i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.41 + 0.998i)T \) |
| 5 | \( 1 + (-2.05 - 0.891i)T \) |
| good | 2 | \( 1 + (-0.199 + 0.0424i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.530 - 0.918i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.73 + 1.21i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.20 - 0.680i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (5.93 - 4.31i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.26 + 1.64i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.406 + 0.450i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.865 - 8.23i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.439 - 4.18i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (3.21 + 9.88i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (6.31 + 1.34i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.32 - 2.29i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.0694 + 0.660i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (5.03 + 3.65i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.348 - 0.0741i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.89 + 1.46i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (1.16 + 11.0i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (3.31 + 2.40i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.32 - 7.14i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.19 + 11.3i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.25 - 2.33i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.63 + 5.04i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0258 - 0.245i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.39805383425495749833660870557, −11.40124971762307871509961137115, −10.59594575682720633428241134936, −9.130541393668006783736090193280, −8.644539389579292454496463711336, −6.91647678038246073238055105305, −6.18737831528035822995317230309, −5.10926966261467730272367024032, −3.72160717456142404466326548263, −1.64042683439239307954911923750,
1.17042111699141903886877279137, 3.99955750102440715885780761272, 4.74812401257303935191640970287, 5.90718968225677281331413206804, 6.65492383807812902070562642949, 8.689645605416081263959605222675, 9.464914984893710365086454288646, 10.01765175357027091164723209685, 11.25247690356011003259446749132, 12.07421288013796355204356186567
