L(s) = 1 | + 1.68e3·11-s + 2.51e6·31-s + 1.38e7·41-s + 9.71e7·61-s + 1.45e8·71-s + 4.37e7·81-s − 1.27e8·101-s − 8.55e8·121-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | + 0.114·11-s + 2.71·31-s + 4.89·41-s + 7.01·61-s + 5.71·71-s + 1.01·81-s − 1.22·101-s − 3.99·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(7.272392762\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.272392762\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.527554746494703816802756249861, −8.136719483830492668842455713557, −7.961945681590457225693885109417, −7.78127429379782835530150506623, −7.63776416526739076503683341092, −6.83189020615364606737820156692, −6.67030960888784217769678152884, −6.56395080067568035436288030926, −6.37331120103137948134377595685, −5.76536379520926824725687042517, −5.30713061590918909546209171180, −5.23777886249820827884923461101, −5.06951152179453231037906655869, −4.18089950912373555953980515566, −4.15443417274065027719482283680, −3.92865559251209604583404900433, −3.57445109686916402073941030506, −2.88214826877039764902045717410, −2.38591889028467664637943897685, −2.36652565405389655473021689192, −2.26416979785946066977442742429, −1.06695379903639156842807263870, −1.00662392204109678114375776006, −0.870903627817817694143726239398, −0.37821866136852964399809706502,
0.37821866136852964399809706502, 0.870903627817817694143726239398, 1.00662392204109678114375776006, 1.06695379903639156842807263870, 2.26416979785946066977442742429, 2.36652565405389655473021689192, 2.38591889028467664637943897685, 2.88214826877039764902045717410, 3.57445109686916402073941030506, 3.92865559251209604583404900433, 4.15443417274065027719482283680, 4.18089950912373555953980515566, 5.06951152179453231037906655869, 5.23777886249820827884923461101, 5.30713061590918909546209171180, 5.76536379520926824725687042517, 6.37331120103137948134377595685, 6.56395080067568035436288030926, 6.67030960888784217769678152884, 6.83189020615364606737820156692, 7.63776416526739076503683341092, 7.78127429379782835530150506623, 7.961945681590457225693885109417, 8.136719483830492668842455713557, 8.527554746494703816802756249861