| L(s) = 1 | + (−11.9 − 10.6i)2-s + (129. + 53.7i)3-s + (29.4 + 254. i)4-s + (832. − 344. i)5-s + (−978. − 2.02e3i)6-s + (1.15e3 + 1.15e3i)7-s + (2.35e3 − 3.35e3i)8-s + (9.33e3 + 9.33e3i)9-s + (−1.36e4 − 4.74e3i)10-s + (−2.30e4 + 9.53e3i)11-s + (−9.85e3 + 3.46e4i)12-s + (1.00e4 + 4.17e3i)13-s + (−1.50e3 − 2.61e4i)14-s + 1.26e5·15-s + (−6.38e4 + 1.49e4i)16-s + 3.72e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.746 − 0.665i)2-s + (1.60 + 0.664i)3-s + (0.114 + 0.993i)4-s + (1.33 − 0.551i)5-s + (−0.755 − 1.56i)6-s + (0.481 + 0.481i)7-s + (0.575 − 0.818i)8-s + (1.42 + 1.42i)9-s + (−1.36 − 0.474i)10-s + (−1.57 + 0.650i)11-s + (−0.475 + 1.66i)12-s + (0.352 + 0.146i)13-s + (−0.0391 − 0.679i)14-s + 2.50·15-s + (−0.973 + 0.228i)16-s + 0.446i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0914i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.995 - 0.0914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.60632 + 0.119476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.60632 + 0.119476i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.9 + 10.6i)T \) |
| good | 3 | \( 1 + (-129. - 53.7i)T + (4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-832. + 344. i)T + (2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-1.15e3 - 1.15e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (2.30e4 - 9.53e3i)T + (1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-1.00e4 - 4.17e3i)T + (5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 3.72e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-7.67e4 + 1.85e5i)T + (-1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (-6.25e4 + 6.25e4i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (2.74e5 - 6.62e5i)T + (-3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 + 8.22e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-3.21e5 + 1.33e5i)T + (2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-1.61e6 - 1.61e6i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (-3.09e6 + 1.28e6i)T + (8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 7.02e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-2.57e6 - 6.21e6i)T + (-4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (8.35e6 + 2.01e7i)T + (-1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (2.83e6 - 6.85e6i)T + (-1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (1.40e7 + 5.80e6i)T + (2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (3.22e7 + 3.22e7i)T + 6.45e14iT^{2} \) |
| 73 | \( 1 + (4.22e6 + 4.22e6i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 3.06e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (1.08e7 - 2.61e7i)T + (-1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-5.47e7 + 5.47e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 + 1.08e8T + 7.83e15T^{2} \) |
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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
−15.06942201217623782391669619760, −13.54967565452080709124941454403, −12.92962470097972504029985888696, −10.70227596375062969356683128923, −9.595920689029616737473060332566, −8.922078849515338376534874406561, −7.77328836511118490638966499868, −4.84439781437214469120447401081, −2.77559435273221817924255158773, −1.86463453535506831458043445522,
1.43564214522503220615047453691, 2.71668365851174535884704026079, 5.79560833451474304731560979190, 7.40252726485851770282468432833, 8.238093672749514499245552213744, 9.592159950657395069687329521182, 10.58215750586339008917322484257, 13.33913814622970949685418255159, 13.92364986002774433122896765830, 14.71631741194166898582422915006
