| L(s) = 1 | − 16·9-s − 208·13-s − 312·29-s − 272·37-s − 96·41-s − 80·49-s − 48·53-s + 1.05e3·61-s − 1.44e3·73-s − 510·81-s − 40·89-s − 4.54e3·97-s − 1.76e3·101-s − 1.69e3·109-s − 64·113-s + 3.32e3·117-s − 332·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.89e4·169-s + ⋯ |
| L(s) = 1 | − 0.592·9-s − 4.43·13-s − 1.99·29-s − 1.20·37-s − 0.365·41-s − 0.233·49-s − 0.124·53-s + 2.21·61-s − 2.30·73-s − 0.699·81-s − 0.0476·89-s − 4.75·97-s − 1.74·101-s − 1.49·109-s − 0.0532·113-s + 2.62·117-s − 0.249·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 8.61·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 766 T^{4} + 16 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 80 T^{2} + 212622 T^{4} + 80 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 332 T^{2} + 474102 T^{4} + 332 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 p T + 6762 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4450 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 2644 T^{2} + 87237846 T^{4} - 2644 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 36816 T^{2} + 602323406 T^{4} + 36816 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 156 T + 33358 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 95612 T^{2} + 4010875782 T^{4} + 95612 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 136 T + 2418 p T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 48 T + 124222 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 66320 T^{2} - 2039763618 T^{4} + 66320 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 204240 T^{2} + 24337141742 T^{4} + 204240 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 24 T + 222298 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 507788 T^{2} + 128506595382 T^{4} + 507788 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 528 T + 462422 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 814480 T^{2} + 336078840702 T^{4} + 814480 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 1385052 T^{2} + 735523082342 T^{4} + 1385052 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 720 T + 745010 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 756668 T^{2} + 456844629702 T^{4} + 756668 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 300176 T^{2} + 429308372286 T^{4} + 300176 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 872438 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1136 T + p^{3} T^{2} )^{4} \) |
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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
−7.42764967466285264250247319530, −7.22539913551722082217566803297, −6.95837542097389002147747889207, −6.89850866348488187587371655092, −6.72332419712555598769501689796, −6.23375153841989479264345590929, −5.95017205605691022085007988502, −5.65313311553355384098826218099, −5.56730194099167278241759052525, −5.20671458341954719046144052941, −5.08321933327784314425046074035, −4.97442389993247159956207949082, −4.55266168912786409725266018039, −4.52141952529205522710938419612, −3.85772264252258717715249302470, −3.77792779277464135182892851701, −3.76684670699694119916689970973, −3.04401721947458475719243641990, −2.71669290027693429690667273492, −2.60778781267707746660457977386, −2.57115108947149486573428068472, −2.01627191104989979402179107199, −1.90195662217019154839614845305, −1.20648376297121202906859645087, −1.20408792002165426340483454154, 0, 0, 0, 0,
1.20408792002165426340483454154, 1.20648376297121202906859645087, 1.90195662217019154839614845305, 2.01627191104989979402179107199, 2.57115108947149486573428068472, 2.60778781267707746660457977386, 2.71669290027693429690667273492, 3.04401721947458475719243641990, 3.76684670699694119916689970973, 3.77792779277464135182892851701, 3.85772264252258717715249302470, 4.52141952529205522710938419612, 4.55266168912786409725266018039, 4.97442389993247159956207949082, 5.08321933327784314425046074035, 5.20671458341954719046144052941, 5.56730194099167278241759052525, 5.65313311553355384098826218099, 5.95017205605691022085007988502, 6.23375153841989479264345590929, 6.72332419712555598769501689796, 6.89850866348488187587371655092, 6.95837542097389002147747889207, 7.22539913551722082217566803297, 7.42764967466285264250247319530
