| L(s) = 1 | + (11.1 − 19.2i)2-s + (31.2 − 18.0i)3-s + (−119. − 207. i)4-s + (−305. − 176. i)5-s − 802. i·6-s + (2.23e3 + 872. i)7-s + 374.·8-s + (−2.63e3 + 4.55e3i)9-s + (−6.79e3 + 3.92e3i)10-s + (6.59e3 + 1.14e4i)11-s + (−7.46e3 − 4.31e3i)12-s − 2.53e4i·13-s + (4.17e4 − 3.33e4i)14-s − 1.27e4·15-s + (3.47e4 − 6.02e4i)16-s + (−1.29e5 + 7.48e4i)17-s + ⋯ |
| L(s) = 1 | + (0.695 − 1.20i)2-s + (0.385 − 0.222i)3-s + (−0.467 − 0.809i)4-s + (−0.488 − 0.281i)5-s − 0.618i·6-s + (0.931 + 0.363i)7-s + 0.0913·8-s + (−0.400 + 0.694i)9-s + (−0.679 + 0.392i)10-s + (0.450 + 0.780i)11-s + (−0.360 − 0.207i)12-s − 0.887i·13-s + (1.08 − 0.869i)14-s − 0.250·15-s + (0.530 − 0.919i)16-s + (−1.55 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0248 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0248 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.45332 - 1.41761i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45332 - 1.41761i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.23e3 - 872. i)T \) |
| good | 2 | \( 1 + (-11.1 + 19.2i)T + (-128 - 221. i)T^{2} \) |
| 3 | \( 1 + (-31.2 + 18.0i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (305. + 176. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-6.59e3 - 1.14e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 2.53e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (1.29e5 - 7.48e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (8.86e4 + 5.12e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (7.99e4 - 1.38e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 6.51e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-6.99e5 + 4.03e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (5.24e5 - 9.08e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.60e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.42e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (7.13e6 + 4.12e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-4.30e6 - 7.45e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (2.41e5 - 1.39e5i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-3.61e6 - 2.08e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.64e7 + 2.85e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 4.27e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-4.46e7 + 2.58e7i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.54e7 - 2.67e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 1.80e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-1.47e7 - 8.51e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.20e8iT - 7.83e15T^{2} \) |
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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
−20.14156178786120784380703828610, −19.54385642355464216780592457941, −17.52678872180755678102932027798, −15.15215525071074901732497407365, −13.55365575743227887908412567291, −12.15892930583629156816224873081, −10.81586185792776373651155814472, −8.219028512551366033306655073424, −4.54603788982598433809380590253, −2.12856305807243819977604489999,
4.28370613629734219039244962429, 6.64476219818561751459379403941, 8.492259709077910615879102951496, 11.37802218862358283455875635480, 13.88367281346851487608685956894, 14.72762868501157905657922480283, 16.03459895028532202416358613820, 17.55178387371222320033562237938, 19.60943574064617884174935602839, 21.22204820717108769570715881789
