L(s) = 1 | + 2.68e8·4-s + 4.67e11·7-s − 7.62e12·9-s + 5.40e16·16-s − 1.49e19·25-s + 1.25e20·28-s − 2.04e21·36-s − 4.71e21·37-s − 4.08e22·43-s + 1.53e23·49-s − 3.56e24·63-s + 9.67e24·64-s − 1.55e25·67-s − 2.56e25·79-s + 5.81e25·81-s − 3.99e27·100-s + 6.70e27·109-s + 2.52e28·112-s + 2.62e28·121-s + 127-s + 131-s + 137-s + 139-s − 4.12e29·144-s − 1.26e30·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 2·4-s + 1.82·7-s − 9-s + 3·16-s − 2·25-s + 3.65·28-s − 2·36-s − 3.18·37-s − 3.62·43-s + 2.33·49-s − 1.82·63-s + 4·64-s − 3.45·67-s − 0.618·79-s + 81-s − 4·100-s + 2.09·109-s + 5.47·112-s + 2·121-s − 3·144-s − 6.36·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(14)\) |
\(\approx\) |
\(4.651190755\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.651190755\) |
\(L(\frac{29}{2})\) |
|
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
−12.23874733454893136068006963820, −11.77578539873234857500530301687, −11.68509462246030437864027379860, −11.03297118142804531556769927621, −10.46872007497625013433709703826, −9.970228839802185168150035925402, −8.501021247232567698251740133826, −8.447450067261896582491769668251, −7.59188568245801507840783088494, −7.20552410411195479258431347769, −6.44336737442072119959139694764, −5.68862987265430938119231365034, −5.37959653283573896230969971547, −4.53769063275819371327239266101, −3.34761987959202979084529110391, −3.18162194803965720185496937340, −1.99140581508136299239510212488, −1.86222731529994638275233278973, −1.52341676774795249843576771581, −0.40251261207247843311409560123,
0.40251261207247843311409560123, 1.52341676774795249843576771581, 1.86222731529994638275233278973, 1.99140581508136299239510212488, 3.18162194803965720185496937340, 3.34761987959202979084529110391, 4.53769063275819371327239266101, 5.37959653283573896230969971547, 5.68862987265430938119231365034, 6.44336737442072119959139694764, 7.20552410411195479258431347769, 7.59188568245801507840783088494, 8.447450067261896582491769668251, 8.501021247232567698251740133826, 9.970228839802185168150035925402, 10.46872007497625013433709703826, 11.03297118142804531556769927621, 11.68509462246030437864027379860, 11.77578539873234857500530301687, 12.23874733454893136068006963820