| L(s) = 1 | + (−5.80 − 0.918i)2-s + (−14.6 − 20.1i)3-s + (−28.0 − 9.11i)4-s + (−3.89 − 1.26i)5-s + (66.3 + 130. i)6-s + (92.8 + 586. i)7-s + (489. + 249. i)8-s + (33.9 − 104. i)9-s + (21.4 + 10.9i)10-s + (−14.7 + 14.7i)11-s + (226. + 698. i)12-s − 1.08e3·13-s − 3.48e3i·14-s + (31.5 + 97.0i)15-s + (−1.08e3 − 785. i)16-s + (−2.53e3 − 4.97e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.725 − 0.114i)2-s + (−0.541 − 0.745i)3-s + (−0.438 − 0.142i)4-s + (−0.0311 − 0.0101i)5-s + (0.307 + 0.602i)6-s + (0.270 + 1.70i)7-s + (0.955 + 0.486i)8-s + (0.0465 − 0.143i)9-s + (0.0214 + 0.0109i)10-s + (−0.0110 + 0.0110i)11-s + (0.131 + 0.404i)12-s − 0.492·13-s − 1.27i·14-s + (0.00934 + 0.0287i)15-s + (−0.264 − 0.191i)16-s + (−0.516 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.608475 - 0.427544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.608475 - 0.427544i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (-2.26e5 - 1.42e4i)T \) |
| good | 2 | \( 1 + (5.80 + 0.918i)T + (60.8 + 19.7i)T^{2} \) |
| 3 | \( 1 + (14.6 + 20.1i)T + (-225. + 693. i)T^{2} \) |
| 5 | \( 1 + (3.89 + 1.26i)T + (1.26e4 + 9.18e3i)T^{2} \) |
| 7 | \( 1 + (-92.8 - 586. i)T + (-1.11e5 + 3.63e4i)T^{2} \) |
| 11 | \( 1 + (14.7 - 14.7i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + 1.08e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (2.53e3 + 4.97e3i)T + (-1.41e7 + 1.95e7i)T^{2} \) |
| 19 | \( 1 + (-3.62e3 - 4.98e3i)T + (-1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 + (-8.99e3 + 4.58e3i)T + (8.70e7 - 1.19e8i)T^{2} \) |
| 29 | \( 1 + (-1.27e4 + 1.27e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + (-4.06e4 + 6.43e3i)T + (8.44e8 - 2.74e8i)T^{2} \) |
| 37 | \( 1 + (-5.90e4 + 9.35e3i)T + (2.44e9 - 7.92e8i)T^{2} \) |
| 41 | \( 1 + (813. - 1.11e3i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + (6.21e4 + 3.16e4i)T + (3.71e9 + 5.11e9i)T^{2} \) |
| 47 | \( 1 - 1.10e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-7.53e3 + 1.47e4i)T + (-1.30e10 - 1.79e10i)T^{2} \) |
| 59 | \( 1 + (-2.09e4 + 1.32e5i)T + (-4.01e10 - 1.30e10i)T^{2} \) |
| 67 | \( 1 + (-1.95e4 - 3.83e4i)T + (-5.31e10 + 7.31e10i)T^{2} \) |
| 71 | \( 1 + (-2.09e5 - 1.06e5i)T + (7.52e10 + 1.03e11i)T^{2} \) |
| 73 | \( 1 + (1.63e5 + 5.04e5i)T + (-1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-3.71e5 + 7.29e5i)T + (-1.42e11 - 1.96e11i)T^{2} \) |
| 83 | \( 1 + (-6.04e5 + 4.39e5i)T + (1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 + (-5.69e4 + 3.59e5i)T + (-4.72e11 - 1.53e11i)T^{2} \) |
| 97 | \( 1 + (-9.66e4 + 1.33e5i)T + (-2.57e11 - 7.92e11i)T^{2} \) |
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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
−13.43193729538642061110855328915, −12.16843991553080726191846005415, −11.56746260413772108915655361361, −9.860675611092551777386175877142, −8.923693938803378095657914573758, −7.77415258866176012122864262233, −6.16581371648015330415877100625, −4.93881036080341568555862615798, −2.24205489436797867442720875438, −0.60116216194372412616518981982,
0.943863072791624523438817443008, 4.00253018378390055897434877287, 4.88465321299776352132269510203, 7.05236017160685365480510447669, 8.081948168986623356682845051274, 9.637381069154199119811268986655, 10.40830307169294665290515759257, 11.23821114271028772341251212318, 13.14321225366463526346700363269, 13.87287479367805402207847265052
