| L(s) = 1 | + (−5.44 − 9.43i)2-s + (97.9 − 169. i)3-s + (196. − 340. i)4-s + (−100. − 173. i)5-s − 2.13e3·6-s − 9.86e3·8-s + (−9.33e3 − 1.61e4i)9-s + (−1.09e3 + 1.89e3i)10-s + (−3.19e4 + 5.53e4i)11-s + (−3.85e4 − 6.67e4i)12-s − 1.64e5·13-s − 3.93e4·15-s + (−4.69e4 − 8.13e4i)16-s + (1.81e5 − 3.14e5i)17-s + (−1.01e5 + 1.76e5i)18-s + (2.18e5 + 3.78e5i)19-s + ⋯ |
| L(s) = 1 | + (−0.240 − 0.416i)2-s + (0.697 − 1.20i)3-s + (0.384 − 0.665i)4-s + (−0.0718 − 0.124i)5-s − 0.671·6-s − 0.851·8-s + (−0.474 − 0.821i)9-s + (−0.0345 + 0.0598i)10-s + (−0.657 + 1.13i)11-s + (−0.536 − 0.928i)12-s − 1.59·13-s − 0.200·15-s + (−0.179 − 0.310i)16-s + (0.526 − 0.912i)17-s + (−0.228 + 0.395i)18-s + (0.384 + 0.665i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.489983 + 0.988413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.489983 + 0.988413i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (5.44 + 9.43i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (-97.9 + 169. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (100. + 173. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (3.19e4 - 5.53e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.64e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.81e5 + 3.14e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.18e5 - 3.78e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (4.59e5 + 7.95e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.68e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (1.73e6 - 3.01e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (9.40e6 + 1.62e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 2.40e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.25e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.77e7 - 4.80e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.63e7 + 8.02e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.26e7 + 2.18e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.46e7 + 6.00e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.16e7 + 2.02e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 1.06e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.05e8 + 1.81e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-7.48e4 - 1.29e5i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 5.21e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (1.49e8 + 2.58e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 8.95e8T + 7.60e17T^{2} \) |
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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
−12.56911607054275040004760645531, −12.12240972073449219113297379338, −10.36987243383678894063518937897, −9.372957175237167296109568909389, −7.72092201505405386712429107995, −6.95353120416222737158720790057, −5.17348249050244173284276165007, −2.64257729739868071303102500115, −1.84887203982069209044533277961, −0.32795095099648580468296351328,
2.72598755066827369895658914748, 3.68227328924098557085593796683, 5.40175156404256909149199361206, 7.29714291625538955303201993726, 8.391144789642605311606394165559, 9.455018600363689200053130922008, 10.65151841200742819114996932645, 11.96112189349952883404458716890, 13.42802672478317891311411813212, 14.90453651556493573309171064487
