| L(s) = 1 | + (23.7 − 41.1i)3-s + (−1.13e3 − 1.96e3i)5-s + (−4.97e3 − 3.95e3i)7-s + (8.71e3 + 1.50e4i)9-s + (−467. + 809. i)11-s − 1.50e5·13-s − 1.08e5·15-s + (−2.87e5 + 4.98e5i)17-s + (−4.91e5 − 8.52e5i)19-s + (−2.80e5 + 1.10e5i)21-s + (3.63e5 + 6.28e5i)23-s + (−1.61e6 + 2.78e6i)25-s + 1.76e6·27-s + 6.19e6·29-s + (4.46e6 − 7.73e6i)31-s + ⋯ |
| L(s) = 1 | + (0.169 − 0.293i)3-s + (−0.813 − 1.40i)5-s + (−0.782 − 0.622i)7-s + (0.442 + 0.766i)9-s + (−0.00962 + 0.0166i)11-s − 1.45·13-s − 0.551·15-s + (−0.835 + 1.44i)17-s + (−0.865 − 1.49i)19-s + (−0.315 + 0.124i)21-s + (0.270 + 0.468i)23-s + (−0.824 + 1.42i)25-s + 0.638·27-s + 1.62·29-s + (0.868 − 1.50i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5340356365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5340356365\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (4.97e3 + 3.95e3i)T \) |
| good | 3 | \( 1 + (-23.7 + 41.1i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.13e3 + 1.96e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (467. - 809. i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.50e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (2.87e5 - 4.98e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (4.91e5 + 8.52e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-3.63e5 - 6.28e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 6.19e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-4.46e6 + 7.73e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (3.70e6 + 6.41e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 5.44e4T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.48e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.37e7 - 2.38e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (4.19e6 - 7.26e6i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.40e6 + 2.42e6i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.77e7 + 3.07e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.93e7 + 3.35e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.46e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (4.26e7 - 7.38e7i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.11e8 - 3.66e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.26e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-5.03e5 - 8.72e5i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 3.88e8T + 7.60e17T^{2} \) |
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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
−12.33334089393041626929604122962, −10.95283774573786271100533789884, −9.778755324477553457457122057257, −8.627681590767731837875088988653, −7.73248245805432032233154363891, −6.66304098829404392788052316989, −4.81657443770479772482079621040, −4.20718708905117068598669421448, −2.36241445472418013773195724268, −0.819819904338279281488838168695,
0.17698183566914278888074389332, 2.56456485366096994412034482499, 3.30244537955770797962659914281, 4.64442008757523428397749499618, 6.54634279315002213237394024824, 7.01337719116710850467738391871, 8.496746108809161426259090391180, 9.815101253791841629595192615622, 10.44010408518326812390436280748, 11.93213882933615791023923028796
