| L(s) = 1 | + (−1.93e4 − 3.34e4i)2-s + (2.39e6 − 4.14e6i)3-s + (−4.77e8 + 8.27e8i)4-s + (−7.79e8 − 1.35e9i)5-s − 1.84e11·6-s + (−1.66e11 − 1.78e12i)7-s + (1.61e13 − 0.00976i)8-s + (−1.14e13 − 1.98e13i)9-s + (−3.01e13 + 5.21e13i)10-s + (3.18e14 − 5.51e14i)11-s + (2.28e15 + 3.95e15i)12-s + 9.87e15·13-s + (−5.65e16 + 4.00e16i)14-s − 7.45e15·15-s + (−5.62e16 − 9.73e16i)16-s + (−5.56e16 + 9.64e16i)17-s + ⋯ |
| L(s) = 1 | + (−0.833 − 1.44i)2-s + (0.288 − 0.499i)3-s + (−0.890 + 1.54i)4-s + (−0.0571 − 0.0989i)5-s − 0.962·6-s + (−0.0929 − 0.995i)7-s + 1.30·8-s + (−0.166 − 0.288i)9-s + (−0.0952 + 0.165i)10-s + (0.252 − 0.437i)11-s + (0.514 + 0.890i)12-s + 0.695·13-s + (−1.36 + 0.964i)14-s − 0.0659·15-s + (−0.195 − 0.337i)16-s + (−0.0801 + 0.138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.7797774535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7797774535\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.39e6 + 4.14e6i)T \) |
| 7 | \( 1 + (1.66e11 + 1.78e12i)T \) |
| good | 2 | \( 1 + (1.93e4 + 3.34e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (7.79e8 + 1.35e9i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-3.18e14 + 5.51e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 - 9.87e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (5.56e16 - 9.64e16i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-4.39e17 - 7.61e17i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (3.00e19 + 5.20e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 5.92e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-1.52e21 + 2.63e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (4.25e22 + 7.37e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 5.72e22T + 5.89e46T^{2} \) |
| 43 | \( 1 + 6.04e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-3.41e23 - 5.92e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (1.35e23 - 2.35e23i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.54e25 + 4.41e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (5.00e25 + 8.66e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (6.46e25 - 1.12e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 8.54e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (1.76e26 - 3.04e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (9.06e26 + 1.56e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 5.87e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-4.97e27 - 8.61e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 6.70e27T + 4.13e57T^{2} \) |
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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
−10.99944528898055934941675401694, −10.12594058218890012278998562063, −8.822273486008695377080420169776, −7.942533962253145188065134364603, −6.41744987409312980047472018214, −4.17020803609285248026109027905, −3.22920486297279251032870300496, −1.99020696340300892739473241538, −0.951324230618102407927612382294, −0.25789645984992404324406453646,
1.44276087682163764044567118992, 3.16252521768267497676748814428, 4.90088187450396636357715477803, 5.93722354486376038888974923644, 7.08083798774831761888623403208, 8.381240579927817228110326324773, 9.096449585760421247139866310881, 10.12682685610259313974331355252, 11.80115499833448165432089336702, 13.63637372684433079741733018291
