L(s) = 1 | − 2.97e7·8-s + 2.34e11·23-s + 4.57e11·25-s + 5.64e11·27-s + 9.96e13·49-s − 1.80e14·59-s + 6.03e14·64-s + 8.92e15·101-s + 1.37e17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 6.98e18·184-s + 191-s + 193-s + 197-s + 199-s − 1.36e19·200-s + ⋯ |
L(s) = 1 | − 1.77·8-s + 3·23-s + 3·25-s + 1.99·27-s + 3·49-s − 1.23·59-s + 2.14·64-s + 0.823·101-s + 3·121-s − 5.31·184-s − 5.31·200-s − 3.54·216-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s+8)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(4.891525886\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.891525886\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.62845953823184035660978521537, −11.91717004391944166157180442578, −11.72502035636706659631663451425, −10.82803332061811798617184074373, −10.80420860657054158737844961865, −10.38729476242867397762562037715, −9.559937044721455388269512012472, −8.897404913890004174101544798843, −8.873758660945888926627681352902, −8.773115665486329559628869883446, −7.87689846616542435416416687270, −6.93079258941494336740045652652, −6.92582463672930799993023009882, −6.56065329641726194264392825412, −5.66635371370094271725637239495, −5.42378459585019875081667253445, −4.68341819350319973420636801256, −4.50038475117504505585998130345, −3.30237062375205593340606069317, −3.05872868649407348423922828523, −2.86621861419539886578318849481, −2.21367948598989338945496235546, −1.09900690167474916942541225891, −0.848513500993516387700676041394, −0.54528391016935919998093648642,
0.54528391016935919998093648642, 0.848513500993516387700676041394, 1.09900690167474916942541225891, 2.21367948598989338945496235546, 2.86621861419539886578318849481, 3.05872868649407348423922828523, 3.30237062375205593340606069317, 4.50038475117504505585998130345, 4.68341819350319973420636801256, 5.42378459585019875081667253445, 5.66635371370094271725637239495, 6.56065329641726194264392825412, 6.92582463672930799993023009882, 6.93079258941494336740045652652, 7.87689846616542435416416687270, 8.773115665486329559628869883446, 8.873758660945888926627681352902, 8.897404913890004174101544798843, 9.559937044721455388269512012472, 10.38729476242867397762562037715, 10.80420860657054158737844961865, 10.82803332061811798617184074373, 11.72502035636706659631663451425, 11.91717004391944166157180442578, 12.62845953823184035660978521537