| L(s) = 1 | + (−3.97 − 0.432i)2-s + (1.94 − 2.28i)3-s + (7.82 + 1.72i)4-s + (6.73 + 5.12i)5-s + (−8.71 + 8.25i)6-s + (−6.58 + 16.5i)7-s + (−0.0516 − 0.0174i)8-s + (−1.45 − 8.88i)9-s + (−24.5 − 23.2i)10-s + (−42.8 + 19.8i)11-s + (19.1 − 14.5i)12-s + (6.69 − 40.8i)13-s + (33.3 − 62.9i)14-s + (24.7 − 5.45i)15-s + (−57.9 − 26.8i)16-s + (20.0 + 50.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.40 − 0.152i)2-s + (0.373 − 0.440i)3-s + (0.978 + 0.215i)4-s + (0.602 + 0.458i)5-s + (−0.593 + 0.561i)6-s + (−0.355 + 0.893i)7-s + (−0.00228 − 0.000769i)8-s + (−0.0539 − 0.328i)9-s + (−0.777 − 0.736i)10-s + (−1.17 + 0.542i)11-s + (0.460 − 0.349i)12-s + (0.142 − 0.871i)13-s + (0.637 − 1.20i)14-s + (0.426 − 0.0939i)15-s + (−0.906 − 0.419i)16-s + (0.285 + 0.717i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.105447 + 0.270229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105447 + 0.270229i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.94 + 2.28i)T \) |
| 59 | \( 1 + (-434. + 129. i)T \) |
| good | 2 | \( 1 + (3.97 + 0.432i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (-6.73 - 5.12i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (6.58 - 16.5i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (42.8 - 19.8i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (-6.69 + 40.8i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (-20.0 - 50.3i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (45.9 + 67.7i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (1.84 - 34.0i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (242. - 26.4i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (121. - 179. i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (-9.02 + 3.04i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (-25.0 - 462. i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (371. + 171. i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (492. - 374. i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (-218. + 207. i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (-368. - 40.1i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (409. + 138. i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (-413. + 314. i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (274. - 518. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-101. - 119. i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (59.2 - 35.6i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (690. - 75.1i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (100. + 190. i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
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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
−12.73253176821293250289475890291, −11.25781645657663631580089690917, −10.30135693671377009897437476294, −9.617758938738550667158690930106, −8.568042802664841623004111428753, −7.80622653718407089703053749826, −6.66219059676543363243203046508, −5.36175988286188658131269124772, −2.87051650933441052161612131428, −1.85132970448794070635907508940,
0.18700495084985941340637133651, 1.93426480994204104100925175240, 3.89377536968040828482732188521, 5.47154831802248360690917607484, 7.03007519391606947486700868784, 7.972849027337333576742020355243, 8.944178827078891325438608926908, 9.756419179554259337804118458273, 10.39932815376650362874134178274, 11.35233323724969478682851142879
