L(s) = 1 | + 4·23-s − 4·43-s + 4·53-s + 4·67-s + 4·107-s − 4·109-s − 2·121-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 + 229-s + ⋯ |
L(s) = 1 | + 4·23-s − 4·43-s + 4·53-s + 4·67-s + 4·107-s − 4·109-s − 2·121-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 + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.390629537\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.390629537\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.94275584401190643759361144731, −6.51609717971453323263356688353, −6.45542394977899089445833295075, −6.23276962458764072875551943722, −6.21576793495913288672666938555, −5.46685489050827273220398100456, −5.37455811743892281392160070934, −5.25395544587937548126499269955, −5.24304145565766964589301061534, −4.93392771874520524762208233616, −4.82380898619976756580846364012, −4.47813848181856758167666029695, −4.16009018173881436333901245907, −3.71071883275013071287590434628, −3.70794689649359819933394860728, −3.50345252333540068860612328111, −3.40043955131161434587863848562, −2.75221736580233919425873058706, −2.69214100945312238063006563296, −2.42588627887330290458188613574, −2.35015299267904367884262980792, −1.70070567771751940877389402323, −1.30402428085307831129897323018, −1.21016810437083424934966732154, −0.72326751539914974162141785262,
0.72326751539914974162141785262, 1.21016810437083424934966732154, 1.30402428085307831129897323018, 1.70070567771751940877389402323, 2.35015299267904367884262980792, 2.42588627887330290458188613574, 2.69214100945312238063006563296, 2.75221736580233919425873058706, 3.40043955131161434587863848562, 3.50345252333540068860612328111, 3.70794689649359819933394860728, 3.71071883275013071287590434628, 4.16009018173881436333901245907, 4.47813848181856758167666029695, 4.82380898619976756580846364012, 4.93392771874520524762208233616, 5.24304145565766964589301061534, 5.25395544587937548126499269955, 5.37455811743892281392160070934, 5.46685489050827273220398100456, 6.21576793495913288672666938555, 6.23276962458764072875551943722, 6.45542394977899089445833295075, 6.51609717971453323263356688353, 6.94275584401190643759361144731