L(s) = 1 | − 4·2-s − 4·3-s + 10·4-s + 4·5-s + 16·6-s + 4·7-s − 20·8-s + 2·9-s − 16·10-s − 40·12-s − 16·14-s − 16·15-s + 35·16-s − 8·17-s − 8·18-s + 12·19-s + 40·20-s − 16·21-s − 8·23-s + 80·24-s + 10·25-s + 16·27-s + 40·28-s + 8·29-s + 64·30-s − 56·32-s + 32·34-s + ⋯ |
L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s + 1.51·7-s − 7.07·8-s + 2/3·9-s − 5.05·10-s − 11.5·12-s − 4.27·14-s − 4.13·15-s + 35/4·16-s − 1.94·17-s − 1.88·18-s + 2.75·19-s + 8.94·20-s − 3.49·21-s − 1.66·23-s + 16.3·24-s + 2·25-s + 3.07·27-s + 7.55·28-s + 1.48·29-s + 11.6·30-s − 9.89·32-s + 5.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.01007046520654578385106043448, −5.57917174414246355419363974273, −5.51222972997683029072480915135, −5.43390181694413172166671518348, −5.24760676318334711282216852906, −5.07615203157050540339042062051, −4.86441559747509794422740720450, −4.78438796642455649214697088900, −4.78430375560844719732710112214, −4.04735172450996448913118132339, −3.97981403744323845693379432029, −3.69357892732632858327948080730, −3.67958962363019632123365297718, −2.98333735214154305490834463400, −2.86482842168831714426382725093, −2.81975094012488672893216423052, −2.81047840927210985558782660770, −2.22320173079674280812291776399, −2.16634126418344892400218322139, −1.96033180228123892551143595526, −1.74331570097576150541923258984, −1.26438023683923859283179244277, −1.24165676357429858224899079803, −1.14898872122668600101275366491, −0.998479830940473430376323975256, 0, 0, 0, 0,
0.998479830940473430376323975256, 1.14898872122668600101275366491, 1.24165676357429858224899079803, 1.26438023683923859283179244277, 1.74331570097576150541923258984, 1.96033180228123892551143595526, 2.16634126418344892400218322139, 2.22320173079674280812291776399, 2.81047840927210985558782660770, 2.81975094012488672893216423052, 2.86482842168831714426382725093, 2.98333735214154305490834463400, 3.67958962363019632123365297718, 3.69357892732632858327948080730, 3.97981403744323845693379432029, 4.04735172450996448913118132339, 4.78430375560844719732710112214, 4.78438796642455649214697088900, 4.86441559747509794422740720450, 5.07615203157050540339042062051, 5.24760676318334711282216852906, 5.43390181694413172166671518348, 5.51222972997683029072480915135, 5.57917174414246355419363974273, 6.01007046520654578385106043448