L(s) = 1 | − 1.32e5i·2-s − 1.17e8i·3-s − 9.07e9·4-s − 1.56e13·6-s + 1.34e13i·7-s + 6.49e13i·8-s − 8.25e15·9-s − 7.96e16·11-s + 1.06e18i·12-s − 1.16e18i·13-s + 1.79e18·14-s − 6.93e19·16-s − 3.27e20i·17-s + 1.09e21i·18-s − 2.21e21·19-s + ⋯ |
L(s) = 1 | − 1.43i·2-s − 1.57i·3-s − 1.05·4-s − 2.26·6-s + 0.153i·7-s + 0.0815i·8-s − 1.48·9-s − 0.522·11-s + 1.66i·12-s − 0.486i·13-s + 0.219·14-s − 0.939·16-s − 1.63i·17-s + 2.12i·18-s − 1.76·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(0.1554568810\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1554568810\) |
\(L(\frac{35}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + 1.32e5iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 1.17e8iT - 5.55e15T^{2} \) |
| 7 | \( 1 - 1.34e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 + 7.96e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.16e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 3.27e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 2.21e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 3.93e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 + 1.59e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 2.07e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 4.76e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 + 6.39e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 2.16e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 + 7.45e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 - 1.70e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 5.91e28T + 2.74e58T^{2} \) |
| 61 | \( 1 - 2.18e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.21e30iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 2.87e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 3.41e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 + 5.87e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 2.11e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 - 2.67e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.07e31iT - 3.65e65T^{2} \) |
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
−10.23653302195385441688767680683, −8.876747193425340039430806824987, −7.61629677366772877574166337947, −6.60570279137269026781164154159, −5.15442806868231077504372258616, −3.47752046297065423527398814332, −2.40329039385924943791021453324, −1.83697122514945304294950225978, −0.69264242345870491518276435380, −0.03868770458918721846629333644,
2.23514204748033604339471587156, 3.92797848348813698030270708406, 4.56789966866343438627753729988, 5.69068246748444769393577832702, 6.62614253619166537709054537862, 8.199037631157385659233435605216, 8.873192635958507949878270032931, 10.23953823405647905559802097126, 11.02191227523356351282135475844, 12.93630398001934286280995887671