| L(s) = 1 | − 3·4-s + 8·7-s − 7·9-s − 10·11-s − 12·17-s + 6·23-s − 24·28-s + 21·36-s − 6·43-s + 30·44-s − 12·47-s + 22·49-s − 20·61-s − 56·63-s + 15·64-s + 36·68-s + 4·73-s − 80·77-s + 20·81-s + 44·83-s − 18·92-s + 70·99-s + 40·101-s − 96·119-s + 41·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 3.02·7-s − 7/3·9-s − 3.01·11-s − 2.91·17-s + 1.25·23-s − 4.53·28-s + 7/2·36-s − 0.914·43-s + 4.52·44-s − 1.75·47-s + 22/7·49-s − 2.56·61-s − 7.05·63-s + 15/8·64-s + 4.36·68-s + 0.468·73-s − 9.11·77-s + 20/9·81-s + 4.82·83-s − 1.87·92-s + 7.03·99-s + 3.98·101-s − 8.80·119-s + 3.72·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + 3 T^{2} + 9 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + 7 T^{2} + 29 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 42 T^{2} + 759 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 31 T^{2} + 721 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 119 T^{2} + 5461 T^{4} + 119 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 63 T^{2} + 2529 T^{4} + 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 114 T^{2} + 6111 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 3 T + 87 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 87 T^{2} + 3729 T^{4} + 87 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 89 T^{2} + 7741 T^{4} - 89 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 67 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 98 T^{2} + 8959 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 114 T^{2} + 6111 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 216 T^{2} + 23646 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 22 T + 282 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 226 T^{2} + 28591 T^{4} + 226 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 363 T^{2} + 51729 T^{4} + 363 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.66734294314587286774730887540, −5.33920381907952844858077269886, −5.12398195367483420549137626344, −5.09611144754851109528460127156, −5.03865197666508716259360995251, −4.91801485489803672976246526582, −4.79966578984389107852094942504, −4.73728253928869061605945092568, −4.65735883141102259682088182730, −4.13932035998118635890942413842, −3.95994281196290031332710496106, −3.87409530151490342028431111903, −3.70505910763620219555140048716, −3.16792405639303256572427844812, −3.03898527746209755051834103371, −3.02898228123381562028295779814, −2.62290792010731926413105816120, −2.55843634352560006244330209877, −2.29186905702727473181062185612, −2.04356528642148507974529970904, −1.90019748627281464900292501401, −1.89382073712651657084662406058, −1.36610378216856527678097345997, −0.906394625779747658877859261218, −0.905640333201684344999824749830, 0, 0, 0, 0,
0.905640333201684344999824749830, 0.906394625779747658877859261218, 1.36610378216856527678097345997, 1.89382073712651657084662406058, 1.90019748627281464900292501401, 2.04356528642148507974529970904, 2.29186905702727473181062185612, 2.55843634352560006244330209877, 2.62290792010731926413105816120, 3.02898228123381562028295779814, 3.03898527746209755051834103371, 3.16792405639303256572427844812, 3.70505910763620219555140048716, 3.87409530151490342028431111903, 3.95994281196290031332710496106, 4.13932035998118635890942413842, 4.65735883141102259682088182730, 4.73728253928869061605945092568, 4.79966578984389107852094942504, 4.91801485489803672976246526582, 5.03865197666508716259360995251, 5.09611144754851109528460127156, 5.12398195367483420549137626344, 5.33920381907952844858077269886, 5.66734294314587286774730887540
