L(s) = 1 | − 2.44·3-s + 10.6i·7-s − 3.00·9-s + 8.71·11-s + 19.5i·13-s − 21.3·17-s − 26.1·19-s − 26.1i·21-s + 10.6i·23-s + 29.3·27-s + 34.8i·29-s − 4i·31-s − 21.3·33-s − 14.6i·37-s − 47.9i·39-s + ⋯ |
L(s) = 1 | − 0.816·3-s + 1.52i·7-s − 0.333·9-s + 0.792·11-s + 1.50i·13-s − 1.25·17-s − 1.37·19-s − 1.24i·21-s + 0.464i·23-s + 1.08·27-s + 1.20i·29-s − 0.129i·31-s − 0.647·33-s − 0.397i·37-s − 1.23i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3320935084\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3320935084\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.44T + 9T^{2} \) |
| 7 | \( 1 - 10.6iT - 49T^{2} \) |
| 11 | \( 1 - 8.71T + 121T^{2} \) |
| 13 | \( 1 - 19.5iT - 169T^{2} \) |
| 17 | \( 1 + 21.3T + 289T^{2} \) |
| 19 | \( 1 + 26.1T + 361T^{2} \) |
| 23 | \( 1 - 10.6iT - 529T^{2} \) |
| 29 | \( 1 - 34.8iT - 841T^{2} \) |
| 31 | \( 1 + 4iT - 961T^{2} \) |
| 37 | \( 1 + 14.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 24T + 1.68e3T^{2} \) |
| 43 | \( 1 - 56.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 10.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 43.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 26.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 7.34T + 4.48e3T^{2} \) |
| 71 | \( 1 + 84iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 100iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 17.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 150T + 7.92e3T^{2} \) |
| 97 | \( 1 - 21.3T + 9.40e3T^{2} \) |
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
−9.419810797143757331836250706234, −8.937384556246484078273449798574, −8.486779618105798649130120976863, −6.95847311188272473045076644637, −6.40773662772113463945563788573, −5.77553661423733099623047088852, −4.84977143198732295273876806566, −4.02090806779794920577767394175, −2.57059714518909647948438369759, −1.73544922519266752793603230483,
0.12345814557368247087006107480, 0.910318445250676894435375568478, 2.50311456572889339952874404994, 3.84156871637260967186478296744, 4.45086791114996261791551578007, 5.47150083944066311405076942662, 6.46056417966862858456251480618, 6.81957548161576430697473318154, 7.985506538415601789864960933501, 8.564055943662387351425179691008