L(s) = 1 | + (−12.6 + 21.9i)2-s + (−7.68 + 4.43i)3-s + (−193. − 334. i)4-s + (−538. − 310. i)5-s − 224. i·6-s + (207. + 2.39e3i)7-s + 3.29e3·8-s + (−3.24e3 + 5.61e3i)9-s + (1.36e4 − 7.87e3i)10-s + (−7.54e3 − 1.30e4i)11-s + (2.96e3 + 1.71e3i)12-s + 4.51e4i·13-s + (−5.51e4 − 2.57e4i)14-s + 5.51e3·15-s + (7.65e3 − 1.32e4i)16-s + (3.59e4 − 2.07e4i)17-s + ⋯ |
L(s) = 1 | + (−0.791 + 1.37i)2-s + (−0.0948 + 0.0547i)3-s + (−0.754 − 1.30i)4-s + (−0.861 − 0.497i)5-s − 0.173i·6-s + (0.0864 + 0.996i)7-s + 0.804·8-s + (−0.494 + 0.855i)9-s + (1.36 − 0.787i)10-s + (−0.515 − 0.892i)11-s + (0.143 + 0.0825i)12-s + 1.57i·13-s + (−1.43 − 0.670i)14-s + 0.108·15-s + (0.116 − 0.202i)16-s + (0.430 − 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0996243 - 0.404143i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0996243 - 0.404143i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-207. - 2.39e3i)T \) |
good | 2 | \( 1 + (12.6 - 21.9i)T + (-128 - 221. i)T^{2} \) |
| 3 | \( 1 + (7.68 - 4.43i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (538. + 310. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (7.54e3 + 1.30e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 4.51e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-3.59e4 + 2.07e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.02e5 - 5.90e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.08e5 - 1.87e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 3.02e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + (1.11e6 - 6.42e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.31e6 + 2.27e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.05e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 6.68e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-8.46e5 - 4.88e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-2.36e5 - 4.08e5i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.62e7 - 9.35e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-2.08e7 - 1.20e7i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-5.58e6 - 9.67e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 7.52e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-8.54e6 + 4.93e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (7.33e6 - 1.27e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 3.82e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-3.43e7 - 1.98e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 7.73e7iT - 7.83e15T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.59762731525319911731407686899, −19.37859080955687407921556806281, −18.35586173558354457926947050254, −16.43503006301631741170493574567, −16.01997352000235736976215571779, −14.26079677183159962009615069588, −11.68657371086322816036152994880, −9.007986940173629443759745171389, −7.81044747441236046957725910062, −5.53847424697892715859035240096,
0.42677902186565970797190865179, 3.36332384515845514557858091343, 7.78651959557278014319826916202, 10.01502500553811457196240400350, 11.25367809818783152389041622299, 12.67606891811590983153740405985, 15.10162438366285117828757996645, 17.41728519699956085515454724301, 18.43815482799568929065553527631, 20.13293929374421321854614412568