L(s) = 1 | + 2·5-s − 2·11-s − 16-s + 3·25-s − 4·55-s − 2·80-s − 4·89-s + 3·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 2·5-s − 2·11-s − 16-s + 3·25-s − 4·55-s − 2·80-s − 4·89-s + 3·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9280014355\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9280014355\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.20527720293515460316132033344, −10.76660777022994655235228169128, −10.47715785806306955454097860720, −10.08336258337823404344411885192, −9.563289442620246560947755463579, −9.413808162322701135677632800148, −8.674904304332766192542308775724, −8.401864268159942187199920064110, −7.83609449669001126086671876843, −7.02468406012674087089570360911, −6.96188246317849590672782469095, −6.18230644516051652951258386661, −5.68195058062788999199735320572, −5.45654632625930193207409868945, −4.82582518091095724664559359195, −4.46091705236518078993507275027, −3.32019965927162093848879516456, −2.56324593771758606772819653651, −2.40520655674812140903479631954, −1.51719217207667209008065103625,
1.51719217207667209008065103625, 2.40520655674812140903479631954, 2.56324593771758606772819653651, 3.32019965927162093848879516456, 4.46091705236518078993507275027, 4.82582518091095724664559359195, 5.45654632625930193207409868945, 5.68195058062788999199735320572, 6.18230644516051652951258386661, 6.96188246317849590672782469095, 7.02468406012674087089570360911, 7.83609449669001126086671876843, 8.401864268159942187199920064110, 8.674904304332766192542308775724, 9.413808162322701135677632800148, 9.563289442620246560947755463579, 10.08336258337823404344411885192, 10.47715785806306955454097860720, 10.76660777022994655235228169128, 11.20527720293515460316132033344