L(s) = 1 | + 1.37e8·4-s + 9.81e14·16-s + 2.19e17·19-s − 7.45e18·25-s − 1.86e20·31-s + 1.31e23·49-s + 1.61e24·61-s − 2.34e24·64-s + 3.02e25·76-s − 1.53e26·79-s − 1.02e27·100-s − 1.07e28·109-s − 2.62e28·121-s − 2.56e28·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.38e30·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 1.02·4-s + 0.0544·16-s + 1.19·19-s − 25-s − 1.37·31-s + 2·49-s + 1.27·61-s − 0.970·64-s + 1.22·76-s − 3.69·79-s − 1.02·100-s − 3.34·109-s − 2·121-s − 1.40·124-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(14)\) |
\(\approx\) |
\(0.7348433960\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7348433960\) |
\(L(\frac{29}{2})\) |
|
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
−11.22635988447275399282232813563, −10.49029826488493496936944323582, −10.18277650680437817899807132702, −9.375365224945746317106420095135, −9.033724963088822378323457485365, −8.257621796461604555474489953675, −7.60301532212992527815415357146, −7.23467243532439341668021457434, −6.80246784577583636968307548890, −6.08658220962331645165265902348, −5.52929081497347232929272021447, −5.21688236131505500082286747159, −4.17429625080559402087087288564, −3.86527519619025042834763129456, −3.11088083300293128658690927042, −2.54733202720512859947311374051, −2.17815853238366495466063728376, −1.37938261532752268918654850577, −1.16605221930568565540667355031, −0.14522626889455851092667612719,
0.14522626889455851092667612719, 1.16605221930568565540667355031, 1.37938261532752268918654850577, 2.17815853238366495466063728376, 2.54733202720512859947311374051, 3.11088083300293128658690927042, 3.86527519619025042834763129456, 4.17429625080559402087087288564, 5.21688236131505500082286747159, 5.52929081497347232929272021447, 6.08658220962331645165265902348, 6.80246784577583636968307548890, 7.23467243532439341668021457434, 7.60301532212992527815415357146, 8.257621796461604555474489953675, 9.033724963088822378323457485365, 9.375365224945746317106420095135, 10.18277650680437817899807132702, 10.49029826488493496936944323582, 11.22635988447275399282232813563