| L(s) = 1 | + (−40.4 + 70.0i)2-s + (−411. + 88.6i)3-s + (−2.25e3 − 3.89e3i)4-s + (−156. − 271. i)5-s + (1.04e4 − 3.24e4i)6-s + (−4.08e4 + 7.07e4i)7-s + 1.98e5·8-s + (1.61e5 − 7.29e4i)9-s + 2.53e4·10-s + (1.63e5 − 2.83e5i)11-s + (1.27e6 + 1.40e6i)12-s + (−2.58e5 − 4.46e5i)13-s + (−3.30e6 − 5.72e6i)14-s + (8.84e4 + 9.76e4i)15-s + (−3.43e6 + 5.94e6i)16-s + 6.16e6·17-s + ⋯ |
| L(s) = 1 | + (−0.894 + 1.54i)2-s + (−0.977 + 0.210i)3-s + (−1.09 − 1.90i)4-s + (−0.0223 − 0.0387i)5-s + (0.547 − 1.70i)6-s + (−0.918 + 1.59i)7-s + 2.14·8-s + (0.911 − 0.412i)9-s + 0.0801·10-s + (0.306 − 0.530i)11-s + (1.47 + 1.62i)12-s + (−0.192 − 0.333i)13-s + (−1.64 − 2.84i)14-s + (0.0300 + 0.0332i)15-s + (−0.818 + 1.41i)16-s + 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.180765 - 0.0248718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.180765 - 0.0248718i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (411. - 88.6i)T \) |
| good | 2 | \( 1 + (40.4 - 70.0i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 5 | \( 1 + (156. + 271. i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (4.08e4 - 7.07e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-1.63e5 + 2.83e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (2.58e5 + 4.46e5i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 - 6.16e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.85e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (1.51e7 + 2.62e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (1.70e7 - 2.95e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (3.19e7 + 5.53e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + 3.37e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (1.73e8 + 3.00e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (4.07e8 - 7.06e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (-5.62e8 + 9.74e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 - 4.18e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (2.15e9 + 3.73e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.80e9 + 3.13e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.29e9 - 2.24e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.08e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.52e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (-4.75e9 + 8.23e9i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-2.27e9 + 3.94e9i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 + 7.34e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-4.91e10 + 8.51e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−18.26924969556163589759050634173, −16.79770125593276377589323773983, −15.98847006663611527025059277208, −14.87749182592661770878832678666, −12.36075277457677847206068562961, −10.05434287801613204006855138304, −8.655357716696892294086846892553, −6.49130920746944859395735582326, −5.52691325230370914064089274124, −0.17273292553401746852148677369,
1.26123273963838522467384278304, 3.88313301668994622194959319066, 7.23336388105815332831154956273, 9.795827095259272069051947527105, 10.68264279203479809808758952530, 12.13219921647435544424481198107, 13.31159418815981977603965220489, 16.68015246291659833014478800946, 17.36028279003058140991351343246, 18.88831434867757300497645589845
