| L(s) = 1 | + (−1.40 + 0.112i)2-s + (0.00307 − 1.73i)3-s + (1.97 − 0.317i)4-s + (2.47 + 0.901i)5-s + (0.190 + 2.44i)6-s + (2.55 + 0.451i)7-s + (−2.74 + 0.670i)8-s + (−2.99 − 0.0106i)9-s + (−3.59 − 0.991i)10-s + (0.556 + 1.52i)11-s + (−0.544 − 3.42i)12-s + (1.88 + 2.24i)13-s + (−3.65 − 0.347i)14-s + (1.56 − 4.28i)15-s + (3.79 − 1.25i)16-s + (−3.28 + 1.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0796i)2-s + (0.00177 − 0.999i)3-s + (0.987 − 0.158i)4-s + (1.10 + 0.403i)5-s + (0.0778 + 0.996i)6-s + (0.967 + 0.170i)7-s + (−0.971 + 0.236i)8-s + (−0.999 − 0.00355i)9-s + (−1.13 − 0.313i)10-s + (0.167 + 0.461i)11-s + (−0.157 − 0.987i)12-s + (0.522 + 0.622i)13-s + (−0.977 − 0.0929i)14-s + (0.405 − 1.10i)15-s + (0.949 − 0.313i)16-s + (−0.797 + 0.460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.977152 - 0.261192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.977152 - 0.261192i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.112i)T \) |
| 3 | \( 1 + (-0.00307 + 1.73i)T \) |
| good | 5 | \( 1 + (-2.47 - 0.901i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-2.55 - 0.451i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.556 - 1.52i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.88 - 2.24i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.28 - 1.89i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.30 + 7.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.07 + 6.06i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.88 - 3.25i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.57 - 0.630i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (6.40 - 3.69i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.43 + 5.28i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.77 - 1.37i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.253 + 1.43i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.180T + 53T^{2} \) |
| 59 | \( 1 + (-0.253 + 0.695i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-11.1 - 1.96i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (10.5 - 8.82i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.57 - 4.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.62 - 4.55i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.72 - 8.01i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.39 + 8.81i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.211 + 0.121i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.95 + 0.711i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.96171565620159159194931773818, −11.22679381356802680134448812135, −10.31833902318951570997512325060, −9.011499462381620008387304336941, −8.451684531553650071020942953686, −7.01343183067809424100003416554, −6.56811753925563711034354737721, −5.25332521090903164975317004740, −2.52395395674829782301623777961, −1.57959733897030530099212198216,
1.66518460925591149989187223347, 3.46388742870077085722902090393, 5.26592069267819956297447818137, 6.03449320859830848370929574804, 7.80140213855877398142622175421, 8.656559205113931287125379939642, 9.564588929354825179354982238793, 10.24025903604799801470498579857, 11.16304993214618202421127221955, 11.93215157906240001292145735048
