L(s) = 1 | + (−9.58 − 9.58i)2-s + (14.5 − 14.5i)3-s + 119. i·4-s + (88.8 − 87.9i)5-s − 279.·6-s + (59.5 + 59.5i)7-s + (535. − 535. i)8-s + 303. i·9-s + (−1.69e3 − 8.50i)10-s + 800.·11-s + (1.74e3 + 1.74e3i)12-s + (−459. + 459. i)13-s − 1.14e3i·14-s + (12.9 − 2.57e3i)15-s − 2.60e3·16-s + (−2.98e3 − 2.98e3i)17-s + ⋯ |
L(s) = 1 | + (−1.19 − 1.19i)2-s + (0.540 − 0.540i)3-s + 1.87i·4-s + (0.710 − 0.703i)5-s − 1.29·6-s + (0.173 + 0.173i)7-s + (1.04 − 1.04i)8-s + 0.416i·9-s + (−1.69 − 0.00850i)10-s + 0.601·11-s + (1.01 + 1.01i)12-s + (−0.209 + 0.209i)13-s − 0.416i·14-s + (0.00383 − 0.764i)15-s − 0.635·16-s + (−0.608 − 0.608i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.478204 - 0.607363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.478204 - 0.607363i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-88.8 + 87.9i)T \) |
good | 2 | \( 1 + (9.58 + 9.58i)T + 64iT^{2} \) |
| 3 | \( 1 + (-14.5 + 14.5i)T - 729iT^{2} \) |
| 7 | \( 1 + (-59.5 - 59.5i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 - 800.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (459. - 459. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (2.98e3 + 2.98e3i)T + 2.41e7iT^{2} \) |
| 19 | \( 1 - 6.97e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (1.20e4 - 1.20e4i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + 2.63e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.61e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (-1.68e4 - 1.68e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 1.24e5T + 4.75e9T^{2} \) |
| 43 | \( 1 + (2.96e4 - 2.96e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (-3.57e4 - 3.57e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + (-4.55e4 + 4.55e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 - 5.23e3iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.00e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + (2.28e5 + 2.28e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 4.71e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (2.42e4 - 2.42e4i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 + 4.76e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (2.96e5 - 2.96e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 - 8.34e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (6.03e4 + 6.03e4i)T + 8.32e11iT^{2} \) |
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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
−21.84453100163284345272516644570, −20.51529701941591028045363896621, −19.48628972706016732929349056213, −18.16893276045428751513998969749, −16.82695360168879587627091321153, −13.61174178667615861659840104661, −11.92082504191767210389529089573, −9.769175835864605702233280154032, −8.284269800758333109038994109596, −1.85368698772543337789492702635,
6.61967743645207152387375531906, 8.851232586766487961827105606129, 10.24420784927499549904886608644, 14.33810302360251065432514538488, 15.39290304387960786084600832959, 17.17094672313422933601882307666, 18.25458548985594139704269206635, 19.96003259981556314722544271414, 21.92626434355142500670309075951, 23.99360758855469315074476198734