| L(s) = 1 | − 4·25-s + 32·37-s − 16·43-s − 32·67-s + 16·79-s − 16·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 4/5·25-s + 5.26·37-s − 2.43·43-s − 3.90·67-s + 1.80·79-s − 1.53·109-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8741042212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8741042212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 14 T^{2} - 165 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 28 T^{2} + 255 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 56 T^{2} + 327 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 74 T^{2} + 1995 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 26 T^{2} - 3045 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 124 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 122 T^{2} + 9555 T^{4} + 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 17 T + p T^{2} )^{4}( 1 + 13 T + p T^{2} )^{4} \) |
| 83 | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.95192292503027058174612881211, −3.93722444898595390940915302071, −3.73286603823492350218274183415, −3.52361645719878237756473310772, −3.41894864157419058418062523361, −3.31435715629010062922099210897, −3.09205411958058174181144189602, −2.89726715847902100194662103997, −2.87980720316947311847233361713, −2.73903612765076397187024135476, −2.68342893309467700032829488682, −2.66527974727772728347979293193, −2.45107058934198002601093787070, −2.12102452005448050368300659858, −2.08494913817402832834375746633, −1.80515744001088338731835586600, −1.79934494615322276085474503671, −1.77524635756871142439363474744, −1.24282247562902839870362250904, −1.16783167790534079494482329370, −1.14372364254687640291933658765, −1.10639221187211929832982516234, −0.59911299969144286664980107820, −0.41538977696104546605999812910, −0.099871221306966092479264999420,
0.099871221306966092479264999420, 0.41538977696104546605999812910, 0.59911299969144286664980107820, 1.10639221187211929832982516234, 1.14372364254687640291933658765, 1.16783167790534079494482329370, 1.24282247562902839870362250904, 1.77524635756871142439363474744, 1.79934494615322276085474503671, 1.80515744001088338731835586600, 2.08494913817402832834375746633, 2.12102452005448050368300659858, 2.45107058934198002601093787070, 2.66527974727772728347979293193, 2.68342893309467700032829488682, 2.73903612765076397187024135476, 2.87980720316947311847233361713, 2.89726715847902100194662103997, 3.09205411958058174181144189602, 3.31435715629010062922099210897, 3.41894864157419058418062523361, 3.52361645719878237756473310772, 3.73286603823492350218274183415, 3.93722444898595390940915302071, 3.95192292503027058174612881211
Plot not available for L-functions of degree greater than 10.
