L(s) = 1 | + 4.85e4·11-s + 7.81e4·16-s − 4.17e6·31-s − 5.28e6·41-s − 5.75e7·61-s − 9.22e7·71-s − 7.87e7·81-s + 7.22e8·101-s + 6.14e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 3.79e9·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 3.31·11-s + 1.19·16-s − 4.51·31-s − 1.87·41-s − 4.15·61-s − 3.63·71-s − 1.82·81-s + 6.94·101-s + 2.86·121-s + 3.95·176-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.019486663\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.019486663\) |
\(L(5)\) |
|
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
−11.59787917445806493331916721336, −10.89270854844029005366075275826, −10.75923517528193821773334966735, −10.33562561084523302323539833859, −9.804950947862602499730862019986, −9.503854729516589760068196366077, −8.940562126313141866949483484668, −8.909358550712252079513390698647, −8.810606409920519879239672108156, −7.950051031464391513647461378696, −7.42269233055240161981700020501, −7.10231960563365664466626627171, −7.02420414343425529715621808214, −6.03751932657559777528337007657, −6.02492577982123154300267029597, −5.74908006895450568341471261491, −4.68859398590647797210003383307, −4.58818484252313081043069292160, −3.65382816190440702852192984635, −3.57094785846168390746734024769, −3.23481496563591788452217803052, −1.90749713065090888877954839981, −1.44747688200536942416589873753, −1.42436558255833054154923093605, −0.30000086782784413153038542258,
0.30000086782784413153038542258, 1.42436558255833054154923093605, 1.44747688200536942416589873753, 1.90749713065090888877954839981, 3.23481496563591788452217803052, 3.57094785846168390746734024769, 3.65382816190440702852192984635, 4.58818484252313081043069292160, 4.68859398590647797210003383307, 5.74908006895450568341471261491, 6.02492577982123154300267029597, 6.03751932657559777528337007657, 7.02420414343425529715621808214, 7.10231960563365664466626627171, 7.42269233055240161981700020501, 7.950051031464391513647461378696, 8.810606409920519879239672108156, 8.909358550712252079513390698647, 8.940562126313141866949483484668, 9.503854729516589760068196366077, 9.804950947862602499730862019986, 10.33562561084523302323539833859, 10.75923517528193821773334966735, 10.89270854844029005366075275826, 11.59787917445806493331916721336