L(s) = 1 | + 6.71e7·4-s + 1.65e12·9-s + 1.68e13·11-s + 3.37e15·16-s + 1.21e16·19-s + 5.42e17·29-s + 8.58e18·31-s + 1.11e20·36-s − 3.67e20·41-s + 1.13e21·44-s + 1.15e21·49-s − 2.61e22·59-s + 1.80e22·61-s + 1.51e23·64-s − 3.84e23·71-s + 8.16e23·76-s + 5.43e23·79-s + 2.02e24·81-s + 3.52e24·89-s + 2.78e25·99-s + 3.72e24·101-s + 9.55e25·109-s + 3.64e25·116-s − 4.02e24·121-s + 5.75e26·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | + 1.99·4-s + 1.95·9-s + 1.61·11-s + 2.99·16-s + 1.26·19-s + 0.284·29-s + 1.95·31-s + 3.90·36-s − 2.54·41-s + 3.23·44-s + 0.861·49-s − 1.91·59-s + 0.869·61-s + 3.99·64-s − 2.78·71-s + 2.52·76-s + 1.03·79-s + 2.82·81-s + 1.51·89-s + 3.16·99-s + 0.329·101-s + 3.25·109-s + 0.569·116-s − 0.0371·121-s + 3.91·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(13)\) |
\(\approx\) |
\(13.21208365\) |
\(L(\frac12)\) |
\(\approx\) |
\(13.21208365\) |
\(L(\frac{27}{2})\) |
|
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
−12.12008900932182728886820474660, −12.03743822382065575559198500553, −11.64666502029399476707814223085, −10.75999455121403711229415765039, −10.03104510430905611961860172574, −10.00899364632405505313022878538, −8.997168779348518884236944855081, −8.050155962195873998427181785333, −7.33676007093481220678349713540, −7.08057749658177703561591762160, −6.42067115473990850310971331411, −6.16228919840969828852366202382, −5.04247148704371041283838971388, −4.35203390620115321510580939190, −3.50987995831921873148194917578, −3.15836819733434756394781710690, −2.20871615100511393397996153632, −1.60779795351867689517147118310, −1.24401322257868636129892952772, −0.839110201010052614100169168926,
0.839110201010052614100169168926, 1.24401322257868636129892952772, 1.60779795351867689517147118310, 2.20871615100511393397996153632, 3.15836819733434756394781710690, 3.50987995831921873148194917578, 4.35203390620115321510580939190, 5.04247148704371041283838971388, 6.16228919840969828852366202382, 6.42067115473990850310971331411, 7.08057749658177703561591762160, 7.33676007093481220678349713540, 8.050155962195873998427181785333, 8.997168779348518884236944855081, 10.00899364632405505313022878538, 10.03104510430905611961860172574, 10.75999455121403711229415765039, 11.64666502029399476707814223085, 12.03743822382065575559198500553, 12.12008900932182728886820474660