L(s) = 1 | + 4·5-s − 16-s − 4·17-s + 10·25-s + 4·37-s − 4·80-s − 16·85-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 4·5-s − 16-s − 4·17-s + 10·25-s + 4·37-s − 4·80-s − 16·85-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.195258527\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.195258527\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.82805233357577585111418683528, −6.70837816087971111233251865946, −6.66352544892860029015835683131, −6.49837795726161038754314142375, −6.16611103227721296195916685166, −6.09118537405788621281013089458, −6.01683665928884551687878272564, −5.55880597242699760921462607712, −5.52083872024425997673813104689, −5.13729643570304754357811602424, −4.96310940970327705418246472101, −4.53352003046742537680602568600, −4.49499505462255821884902086408, −4.38529663055047805334252454283, −4.28167444796363657382045257402, −3.49025780032061954018839079916, −3.34491376570338424905121438394, −2.85456487422485691404274893742, −2.48777114432843828978640099774, −2.33572953309973904024194901449, −2.33040060687394007776341711173, −2.22855267312832248037897920716, −1.75586070676143837343040325415, −1.19643705564035640918987382617, −1.12183159499626468381880677725,
1.12183159499626468381880677725, 1.19643705564035640918987382617, 1.75586070676143837343040325415, 2.22855267312832248037897920716, 2.33040060687394007776341711173, 2.33572953309973904024194901449, 2.48777114432843828978640099774, 2.85456487422485691404274893742, 3.34491376570338424905121438394, 3.49025780032061954018839079916, 4.28167444796363657382045257402, 4.38529663055047805334252454283, 4.49499505462255821884902086408, 4.53352003046742537680602568600, 4.96310940970327705418246472101, 5.13729643570304754357811602424, 5.52083872024425997673813104689, 5.55880597242699760921462607712, 6.01683665928884551687878272564, 6.09118537405788621281013089458, 6.16611103227721296195916685166, 6.49837795726161038754314142375, 6.66352544892860029015835683131, 6.70837816087971111233251865946, 6.82805233357577585111418683528