L(s) = 1 | + 2·13-s − 2·17-s + 4·29-s + 2·37-s − 2·53-s + 2·73-s − 81-s + 2·97-s + 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 2·13-s − 2·17-s + 4·29-s + 2·37-s − 2·53-s + 2·73-s − 81-s + 2·97-s + 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.641697496\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.641697496\) |
\(L(1)\) |
|
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
−8.898343517022490814201378567211, −8.630053899975606689695176076637, −8.253870150607420174793273787104, −8.186138585564438907729875065059, −7.61763891168823077212505296624, −7.06396529055362152126001760228, −6.58140971038435988410035407164, −6.45098444084630268463399899097, −6.06862884331546124309552301122, −5.95113047459660457500579286515, −4.98006158610815547455180929820, −4.78761345212855034210472940620, −4.38038051118067235606833245271, −4.13953403024065277636310202923, −3.29244737107063561412180363178, −3.24224194484881919221777398388, −2.41924396642072560160029415616, −2.22088826751870483828544639037, −1.25243281903463886346800228200, −0.914987420435682091305445612252,
0.914987420435682091305445612252, 1.25243281903463886346800228200, 2.22088826751870483828544639037, 2.41924396642072560160029415616, 3.24224194484881919221777398388, 3.29244737107063561412180363178, 4.13953403024065277636310202923, 4.38038051118067235606833245271, 4.78761345212855034210472940620, 4.98006158610815547455180929820, 5.95113047459660457500579286515, 6.06862884331546124309552301122, 6.45098444084630268463399899097, 6.58140971038435988410035407164, 7.06396529055362152126001760228, 7.61763891168823077212505296624, 8.186138585564438907729875065059, 8.253870150607420174793273787104, 8.630053899975606689695176076637, 8.898343517022490814201378567211