| 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
−12.07421288013796355204356186567, −11.25247690356011003259446749132, −10.01765175357027091164723209685, −9.464914984893710365086454288646, −8.689645605416081263959605222675, −6.65492383807812902070562642949, −5.90718968225677281331413206804, −4.74812401257303935191640970287, −3.99955750102440715885780761272, −1.17042111699141903886877279137,
1.64042683439239307954911923750, 3.72160717456142404466326548263, 5.10926966261467730272367024032, 6.18737831528035822995317230309, 6.91647678038246073238055105305, 8.644539389579292454496463711336, 9.130541393668006783736090193280, 10.59594575682720633428241134936, 11.40124971762307871509961137115, 12.39805383425495749833660870557
