| L(s) = 1 | + 122·4-s − 2.52e3·7-s + 1.91e4·13-s − 1.50e3·16-s − 4.38e4·19-s − 5.05e3·25-s − 3.07e5·28-s − 1.01e5·31-s + 4.92e5·37-s + 6.31e5·43-s + 3.12e6·49-s + 2.33e6·52-s − 9.95e5·61-s − 2.18e6·64-s + 2.67e6·67-s + 6.50e6·73-s − 5.35e6·76-s + 1.21e7·79-s − 4.83e7·91-s + 1.31e7·97-s − 6.16e5·100-s + 8.25e5·103-s + 3.69e7·109-s + 3.78e6·112-s − 3.67e7·121-s − 1.24e7·124-s + 127-s + ⋯ |
| L(s) = 1 | + 0.953·4-s − 2.77·7-s + 2.41·13-s − 0.0915·16-s − 1.46·19-s − 0.0646·25-s − 2.64·28-s − 0.613·31-s + 1.59·37-s + 1.21·43-s + 3.79·49-s + 2.30·52-s − 0.561·61-s − 1.04·64-s + 1.08·67-s + 1.95·73-s − 1.39·76-s + 2.77·79-s − 6.72·91-s + 1.46·97-s − 0.0616·100-s + 0.0743·103-s + 2.73·109-s + 0.254·112-s − 1.88·121-s − 0.585·124-s + 4.07·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.746349880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746349880\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 61 p T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 202 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 1261 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 36791014 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 737 p T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 369925954 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 21931 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 575108306 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 33453109930 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 50908 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 246467 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 16245728434 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 315512 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 832483078654 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2333195229562 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4047812406406 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 497953 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 1336361 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 17376702039790 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 3250793 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6075485 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12831979431434 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 81596562001742 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6570629 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.09573076286654212437289346155, −15.54151922671982725401025626497, −15.30531053491726731468585618153, −14.06635388479048649217379819389, −13.35423197678409463548671907688, −12.87426377964223716994391174364, −12.58329156384406705025285111853, −11.46009455567208780498189802363, −10.86278126331819736099924331314, −10.35893280379392073997963270173, −9.358301319847822476760555356046, −8.986975780076536848194280762084, −7.87372807022519824818906293427, −6.61824225810364026712943063372, −6.42555123176556710940437182330, −5.95823610460272958072634246998, −3.93866050416968271871970749261, −3.33661530301984159207579782610, −2.29744490774972589050450996139, −0.65237155630376071661756896066,
0.65237155630376071661756896066, 2.29744490774972589050450996139, 3.33661530301984159207579782610, 3.93866050416968271871970749261, 5.95823610460272958072634246998, 6.42555123176556710940437182330, 6.61824225810364026712943063372, 7.87372807022519824818906293427, 8.986975780076536848194280762084, 9.358301319847822476760555356046, 10.35893280379392073997963270173, 10.86278126331819736099924331314, 11.46009455567208780498189802363, 12.58329156384406705025285111853, 12.87426377964223716994391174364, 13.35423197678409463548671907688, 14.06635388479048649217379819389, 15.30531053491726731468585618153, 15.54151922671982725401025626497, 16.09573076286654212437289346155
