| L(s) = 1 | + (14.7 + 6.24i)2-s + (14.2 + 34.3i)3-s + (177. + 184. i)4-s + (−33.8 + 81.7i)5-s + (−5.04 + 594. i)6-s + (−2.38e3 + 2.38e3i)7-s + (1.47e3 + 3.82e3i)8-s + (3.66e3 − 3.66e3i)9-s + (−1.01e3 + 993. i)10-s + (−4.48e3 + 1.08e4i)11-s + (−3.79e3 + 8.72e3i)12-s + (−1.84e3 − 4.46e3i)13-s + (−4.99e4 + 2.02e4i)14-s − 3.29e3·15-s + (−2.22e3 + 6.54e4i)16-s + 5.26e4i·17-s + ⋯ |
| L(s) = 1 | + (0.920 + 0.390i)2-s + (0.175 + 0.423i)3-s + (0.695 + 0.718i)4-s + (−0.0542 + 0.130i)5-s + (−0.00389 + 0.458i)6-s + (−0.992 + 0.992i)7-s + (0.359 + 0.933i)8-s + (0.558 − 0.558i)9-s + (−0.101 + 0.0993i)10-s + (−0.306 + 0.738i)11-s + (−0.182 + 0.420i)12-s + (−0.0647 − 0.156i)13-s + (−1.30 + 0.526i)14-s − 0.0650·15-s + (−0.0339 + 0.999i)16-s + 0.629i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.39949 + 2.48980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39949 + 2.48980i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-14.7 - 6.24i)T \) |
| good | 3 | \( 1 + (-14.2 - 34.3i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (33.8 - 81.7i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (2.38e3 - 2.38e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (4.48e3 - 1.08e4i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (1.84e3 + 4.46e3i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 5.26e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-8.54e3 + 3.54e3i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (1.68e5 + 1.68e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (-6.06e5 + 2.51e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 4.14e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-5.32e5 + 1.28e6i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-3.86e6 + 3.86e6i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (-1.30e6 + 3.15e6i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 - 8.55e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (4.43e6 + 1.83e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (-1.90e6 - 7.88e5i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (9.82e6 - 4.06e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-1.21e7 - 2.92e7i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (1.49e7 - 1.49e7i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (4.54e6 - 4.54e6i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 4.02e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (5.09e6 - 2.10e6i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (5.75e7 + 5.75e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 - 9.29e7T + 7.83e15T^{2} \) |
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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
−15.48539287351836525321042970930, −14.44870991969733073775363837728, −12.78153513326250480206777384085, −12.26916909281746676737539468493, −10.38513666625403808449638147403, −8.924184608906993488757917042985, −7.09209597643052382657691310737, −5.78847880429720761899596009870, −4.09888062703093035125534818921, −2.63180328863110322692834672182,
0.915469050132830626365338749892, 2.89134305434214703749595469646, 4.46127206311750316136636516491, 6.32049328486424611798913845929, 7.56849767023454418414591257433, 9.825962183337950925199802660157, 10.92593216260944301813539305066, 12.46561288584898338181728654265, 13.42112795504617386388347012992, 14.06854257994364275451446025251
