L(s) = 1 | + (65.8 − 62.0i)2-s + 1.23e3i·3-s + (489. − 8.17e3i)4-s − 2.52e4i·5-s + (7.64e4 + 8.11e4i)6-s + 6.08e5·7-s + (−4.75e5 − 5.69e5i)8-s + 7.79e4·9-s + (−1.56e6 − 1.66e6i)10-s − 7.12e6i·11-s + (1.00e7 + 6.02e5i)12-s + 1.12e7i·13-s + (4.00e7 − 3.77e7i)14-s + 3.11e7·15-s + (−6.66e7 − 8.00e6i)16-s − 5.09e7·17-s + ⋯ |
L(s) = 1 | + (0.727 − 0.685i)2-s + 0.975i·3-s + (0.0597 − 0.998i)4-s − 0.723i·5-s + (0.668 + 0.709i)6-s + 1.95·7-s + (−0.640 − 0.767i)8-s + 0.0488·9-s + (−0.495 − 0.526i)10-s − 1.21i·11-s + (0.973 + 0.0582i)12-s + 0.648i·13-s + (1.42 − 1.33i)14-s + 0.705·15-s + (−0.992 − 0.119i)16-s − 0.511·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(2.55512 - 1.19527i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.55512 - 1.19527i\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-65.8 + 62.0i)T \) |
good | 3 | \( 1 - 1.23e3iT - 1.59e6T^{2} \) |
| 5 | \( 1 + 2.52e4iT - 1.22e9T^{2} \) |
| 7 | \( 1 - 6.08e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 7.12e6iT - 3.45e13T^{2} \) |
| 13 | \( 1 - 1.12e7iT - 3.02e14T^{2} \) |
| 17 | \( 1 + 5.09e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.86e8iT - 4.20e16T^{2} \) |
| 23 | \( 1 + 3.07e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.74e8iT - 1.02e19T^{2} \) |
| 31 | \( 1 + 2.41e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.55e10iT - 2.43e20T^{2} \) |
| 41 | \( 1 + 2.32e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 6.55e10iT - 1.71e21T^{2} \) |
| 47 | \( 1 + 6.84e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 9.64e10iT - 2.60e22T^{2} \) |
| 59 | \( 1 - 3.22e11iT - 1.04e23T^{2} \) |
| 61 | \( 1 - 6.66e11iT - 1.61e23T^{2} \) |
| 67 | \( 1 + 5.91e11iT - 5.48e23T^{2} \) |
| 71 | \( 1 - 8.43e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 7.21e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 5.78e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.54e11iT - 8.87e24T^{2} \) |
| 89 | \( 1 - 2.38e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.26e13T + 6.73e25T^{2} \) |
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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
−18.52347794703156014354139906089, −16.55995466000176649178516657766, −15.02879127992900801465866422669, −13.81902721714609421196886197479, −11.78285288767715316408471987830, −10.63564263509288000676105637554, −8.727530566140184871235034888812, −5.23353068686638011330789676137, −4.13847733670970732735505770859, −1.47206363951694865860601046652,
2.05597772219470668917871605546, 4.79021458568723709447934504244, 6.95079489358404702701397500004, 7.977332520532549214505395441524, 11.28660286471362163711734934589, 12.76940851115830205490987300544, 14.29071813278238700296139301621, 15.22840407429498477697154768548, 17.71589917394948224495098286825, 18.01190991297890747425172424913