| 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
−21.22204820717108769570715881789, −19.60943574064617884174935602839, −17.55178387371222320033562237938, −16.03459895028532202416358613820, −14.72762868501157905657922480283, −13.88367281346851487608685956894, −11.37802218862358283455875635480, −8.492259709077910615879102951496, −6.64476219818561751459379403941, −4.28370613629734219039244962429,
2.12856305807243819977604489999, 4.54603788982598433809380590253, 8.219028512551366033306655073424, 10.81586185792776373651155814472, 12.15892930583629156816224873081, 13.55365575743227887908412567291, 15.15215525071074901732497407365, 17.52678872180755678102932027798, 19.54385642355464216780592457941, 20.14156178786120784380703828610
