| L(s) = 1 | + (−16 + 27.7i)2-s + (265. + 459. i)3-s + (−511. − 886. i)4-s + (6.62e3 − 1.14e4i)5-s − 1.69e4·6-s + (4.34e3 − 4.42e4i)7-s + 3.27e4·8-s + (−5.20e4 + 9.01e4i)9-s + (2.11e5 + 3.67e5i)10-s + (−2.68e5 − 4.64e5i)11-s + (2.71e5 − 4.70e5i)12-s + 1.10e6·13-s + (1.15e6 + 8.28e5i)14-s + 7.02e6·15-s + (−5.24e5 + 9.08e5i)16-s + (4.58e5 + 7.93e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.630 + 1.09i)3-s + (−0.249 − 0.433i)4-s + (0.947 − 1.64i)5-s − 0.891·6-s + (0.0977 − 0.995i)7-s + 0.353·8-s + (−0.293 + 0.509i)9-s + (0.670 + 1.16i)10-s + (−0.502 − 0.870i)11-s + (0.315 − 0.545i)12-s + 0.826·13-s + (0.574 + 0.411i)14-s + 2.38·15-s + (−0.125 + 0.216i)16-s + (0.0782 + 0.135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0344i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.97597 - 0.0340744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97597 - 0.0340744i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 - 27.7i)T \) |
| 7 | \( 1 + (-4.34e3 + 4.42e4i)T \) |
| good | 3 | \( 1 + (-265. - 459. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (-6.62e3 + 1.14e4i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (2.68e5 + 4.64e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.10e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-4.58e5 - 7.93e5i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.86e6 - 3.22e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.12e7 - 1.95e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.08e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.25e8 - 2.17e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.01e8 + 3.49e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.26e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.92e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-5.04e8 + 8.74e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (2.26e8 + 3.91e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.81e9 - 3.14e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (4.93e9 - 8.55e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-9.55e9 - 1.65e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 5.71e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (4.81e9 + 8.34e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (7.83e9 - 1.35e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 1.10e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.43e10 + 4.21e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.33e11T + 7.15e21T^{2} \) |
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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
−16.52082213629641010721432403423, −15.97234423212235115704604559800, −14.14527968830051605090132264201, −13.19736660080944441636167979638, −10.44937152761077249059505720062, −9.280282514171800091017252143758, −8.298350773915352451704261050097, −5.62387379605224122958712243577, −4.14005260317639810023622168779, −1.03687875556164256421848047273,
1.98132651661050457366425651786, 2.70651237357779342933101771980, 6.32693243609154122973787998788, 7.80694386158115997859180123285, 9.597290281683007872127067001489, 11.08045225881733555912863957246, 12.78297411073900438835415617317, 13.89567344869995829010283792252, 15.12449396870903749565165998469, 17.73913768162194200180144405717
