| L(s) = 1 | + 8.15i·2-s − 11.7i·3-s + 189.·4-s + 1.15e3i·5-s + 96.1·6-s − 1.56e3i·7-s + 3.63e3i·8-s + 6.42e3·9-s − 9.40e3·10-s − 1.00e4·11-s − 2.23e3i·12-s − 1.63e3·13-s + 1.27e4·14-s + 1.35e4·15-s + 1.88e4·16-s + 2.59e4·17-s + ⋯ |
| L(s) = 1 | + 0.509i·2-s − 0.145i·3-s + 0.740·4-s + 1.84i·5-s + 0.0741·6-s − 0.652i·7-s + 0.886i·8-s + 0.978·9-s − 0.940·10-s − 0.686·11-s − 0.107i·12-s − 0.0572·13-s + 0.332·14-s + 0.268·15-s + 0.288·16-s + 0.310·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.972190 + 1.89896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.972190 + 1.89896i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.99e6 + 2.77e6i)T \) |
| good | 2 | \( 1 - 8.15iT - 256T^{2} \) |
| 3 | \( 1 + 11.7iT - 6.56e3T^{2} \) |
| 5 | \( 1 - 1.15e3iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.56e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.00e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 1.63e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 2.59e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.67e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 2.25e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 8.29e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 8.03e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.67e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 2.02e6T + 7.98e12T^{2} \) |
| 47 | \( 1 - 6.67e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.35e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.38e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + 2.78e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 2.07e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.64e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.21e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 2.21e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 1.20e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 4.53e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.49e8T + 7.83e15T^{2} \) |
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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
−14.72259828748848250972529832571, −13.84058686291442840150452439438, −12.14928661713604717806557029633, −10.66807947448385134809183866166, −10.30261889608318417012157450632, −7.63418544598004921177890186630, −7.15426180277577005700895082028, −5.95490803053160597114769898111, −3.54173885904885368592389439794, −2.02721087124314925517993233463,
0.796167258052677394494734866817, 2.17711096856830047305012413966, 4.29246333975025535904929526341, 5.65368385292609410827341514276, 7.58195208792194325836416681439, 9.012052751770682896553351957597, 10.08802872482619592403450819190, 11.64814129423278266965610106205, 12.56845506812636978014544448729, 13.24026514677935958993340182385
