| 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 | \( 1 - 21.3T + 9.40e3T^{2} \) |
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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
−8.564055943662387351425179691008, −7.985506538415601789864960933501, −6.81957548161576430697473318154, −6.46056417966862858456251480618, −5.47150083944066311405076942662, −4.45086791114996261791551578007, −3.84156871637260967186478296744, −2.50311456572889339952874404994, −0.910318445250676894435375568478, −0.12345814557368247087006107480,
1.73544922519266752793603230483, 2.57059714518909647948438369759, 4.02090806779794920577767394175, 4.84977143198732295273876806566, 5.77553661423733099623047088852, 6.40773662772113463945563788573, 6.95847311188272473045076644637, 8.486779618105798649130120976863, 8.937384556246484078273449798574, 9.419810797143757331836250706234
