| L(s) = 1 | + 8·2-s + 36·4-s + 120·8-s + 8·11-s + 330·16-s + 64·22-s − 8·23-s + 4·25-s + 16·29-s + 792·32-s − 16·37-s − 8·43-s + 288·44-s − 64·46-s + 32·50-s − 16·53-s + 128·58-s + 1.71e3·64-s + 72·67-s + 48·71-s − 128·74-s + 16·79-s + 9·81-s − 64·86-s + 960·88-s − 288·92-s + 144·100-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s + 42.4·8-s + 2.41·11-s + 82.5·16-s + 13.6·22-s − 1.66·23-s + 4/5·25-s + 2.97·29-s + 140.·32-s − 2.63·37-s − 1.21·43-s + 43.4·44-s − 9.43·46-s + 4.52·50-s − 2.19·53-s + 16.8·58-s + 214.5·64-s + 8.79·67-s + 5.69·71-s − 14.8·74-s + 1.80·79-s + 81-s − 6.90·86-s + 102.·88-s − 30.0·92-s + 72/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(836.9101097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(836.9101097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T^{2} + 2 p T^{4} + 176 T^{6} - 989 T^{8} + 176 p^{2} T^{10} + 2 p^{5} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 4 T - 7 T^{2} - 4 T^{3} + 232 T^{4} - 4 p T^{5} - 7 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 4 T^{2} + 106 T^{4} + 1712 T^{6} - 23165 T^{8} + 1712 p^{2} T^{10} + 106 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 8 T^{2} - 455 T^{4} - 472 T^{6} + 167344 T^{8} - 472 p^{2} T^{10} - 455 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 48 T^{2} + 1033 T^{4} - 26352 T^{6} + 653376 T^{8} - 26352 p^{2} T^{10} + 1033 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 4 T - 22 T^{2} - 32 T^{3} + 547 T^{4} - 32 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 76 T^{2} + 2934 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8 T - 14 T^{2} + 32 T^{3} + 2347 T^{4} + 32 p T^{5} - 14 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 88 T^{2} + 3769 T^{4} - 53944 T^{6} - 42800 T^{8} - 53944 p^{2} T^{10} + 3769 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 4 T - 71 T^{2} + 4 T^{3} + 5032 T^{4} + 4 p T^{5} - 71 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 76 T^{2} + 2790 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 8 T - 10 T^{2} - 256 T^{3} - 725 T^{4} - 256 p T^{5} - 10 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 160 T^{2} + 12039 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 36 T^{2} + 4694 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 18 T + 203 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 216 T^{2} + 24409 T^{4} - 2503224 T^{6} + 217900944 T^{8} - 2503224 p^{2} T^{10} + 24409 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 32 T^{2} + 6190 T^{4} - 606208 T^{6} - 27825101 T^{8} - 606208 p^{2} T^{10} + 6190 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 24 T^{2} + 3289 T^{4} + 516744 T^{6} - 86588496 T^{8} + 516744 p^{2} T^{10} + 3289 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.44841249142198992604497987155, −4.18699049159293504617610676629, −4.05353120972940301898096031622, −3.84838826292867899419254302013, −3.81636229255907107305742194651, −3.72806045049423439004295266105, −3.68651213930141926315831767197, −3.65361718003387974963259365383, −3.25960743646712911531042217440, −3.24896514804327070167305097754, −3.21637288951050862899342757740, −3.19067462473691864374738073886, −3.05491172677448435731186771378, −2.44759619915682041507442755616, −2.40232177895859197406156903881, −2.23529630998355580012508814865, −2.21394590791337583952245270181, −2.05326011883508828501889527970, −2.04368538913935319768105304537, −2.01059945447962417117657110988, −1.48543453054798297276431184004, −1.18370013757149076263716823460, −1.02547229819349956648295903455, −0.992516637622492011154162764369, −0.67189069398895597918953258904,
0.67189069398895597918953258904, 0.992516637622492011154162764369, 1.02547229819349956648295903455, 1.18370013757149076263716823460, 1.48543453054798297276431184004, 2.01059945447962417117657110988, 2.04368538913935319768105304537, 2.05326011883508828501889527970, 2.21394590791337583952245270181, 2.23529630998355580012508814865, 2.40232177895859197406156903881, 2.44759619915682041507442755616, 3.05491172677448435731186771378, 3.19067462473691864374738073886, 3.21637288951050862899342757740, 3.24896514804327070167305097754, 3.25960743646712911531042217440, 3.65361718003387974963259365383, 3.68651213930141926315831767197, 3.72806045049423439004295266105, 3.81636229255907107305742194651, 3.84838826292867899419254302013, 4.05353120972940301898096031622, 4.18699049159293504617610676629, 4.44841249142198992604497987155
Plot not available for L-functions of degree greater than 10.
