| L(s) = 1 | + (−0.109 − 0.480i)2-s + (0.181 − 0.461i)3-s + (1.58 − 0.762i)4-s + (−0.911 + 2.32i)5-s + (−0.241 − 0.0364i)6-s + (−1.18 + 2.36i)7-s + (−1.15 − 1.44i)8-s + (2.01 + 1.87i)9-s + (1.21 + 0.183i)10-s + (−1.01 + 0.937i)11-s + (−0.0651 − 0.869i)12-s + (−1.17 + 3.40i)13-s + (1.26 + 0.311i)14-s + (0.907 + 0.842i)15-s + (1.62 − 2.03i)16-s + (−2.94 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.0775 − 0.339i)2-s + (0.104 − 0.266i)3-s + (0.791 − 0.381i)4-s + (−0.407 + 1.03i)5-s + (−0.0987 − 0.0148i)6-s + (−0.448 + 0.893i)7-s + (−0.408 − 0.512i)8-s + (0.672 + 0.624i)9-s + (0.384 + 0.0579i)10-s + (−0.304 + 0.282i)11-s + (−0.0188 − 0.250i)12-s + (−0.326 + 0.945i)13-s + (0.338 + 0.0832i)14-s + (0.234 + 0.217i)15-s + (0.405 − 0.508i)16-s + (−0.714 − 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23459 + 0.657948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23459 + 0.657948i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.18 - 2.36i)T \) |
| 13 | \( 1 + (1.17 - 3.40i)T \) |
| good | 2 | \( 1 + (0.109 + 0.480i)T + (-1.80 + 0.867i)T^{2} \) |
| 3 | \( 1 + (-0.181 + 0.461i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (0.911 - 2.32i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.01 - 0.937i)T + (0.822 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.94 + 1.41i)T + (10.5 + 13.2i)T^{2} \) |
| 19 | \( 1 + (4.07 - 7.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.25 + 2.04i)T + (14.3 - 17.9i)T^{2} \) |
| 29 | \( 1 + (2.11 - 1.44i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (0.961 - 1.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.590 - 0.284i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (-5.91 + 0.891i)T + (39.1 - 12.0i)T^{2} \) |
| 43 | \( 1 + (-3.01 - 0.454i)T + (41.0 + 12.6i)T^{2} \) |
| 47 | \( 1 + (-6.62 + 6.15i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.308 + 4.11i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-6.23 + 7.82i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (1.02 - 0.701i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (2.23 + 3.86i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.1 - 7.59i)T + (25.9 + 66.0i)T^{2} \) |
| 73 | \( 1 + (-1.79 - 1.66i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-7.16 - 12.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.03 + 8.91i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (2.90 - 12.7i)T + (-80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (3.55 + 6.15i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.77085991914790546831989783354, −10.06878217799936769287747275535, −9.153783337109305993357232912140, −7.929914270382028066656030844255, −6.90474453595975482408109363571, −6.62216465091189479874784292408, −5.36233708146939249745877734367, −3.90514098793711292180451371764, −2.60085385671892910797255355169, −1.98129437668217230004028693619,
0.74495285702328672780021617897, 2.71285600849507027618394667961, 3.91118979774946904687526818165, 4.76505297697183754966625545331, 6.08921469173136105297313750986, 7.04962465219793265235459324633, 7.67960116738929064187433431820, 8.707995978507890833027827823205, 9.373359183155645380243883371536, 10.63810553926728499820780861817
