| L(s) = 1 | + 8·2-s + 36·4-s + 120·8-s − 8·9-s + 330·16-s − 64·18-s − 8·23-s − 4·25-s + 792·32-s − 288·36-s − 64·46-s − 32·50-s + 40·53-s + 1.71e3·64-s − 960·72-s + 48·79-s + 30·81-s − 288·92-s − 144·100-s + 320·106-s − 32·109-s + 96·113-s + 44·121-s + 127-s + 3.43e3·128-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s + 42.4·8-s − 8/3·9-s + 82.5·16-s − 15.0·18-s − 1.66·23-s − 4/5·25-s + 140.·32-s − 48·36-s − 9.43·46-s − 4.52·50-s + 5.49·53-s + 214.5·64-s − 113.·72-s + 5.40·79-s + 10/3·81-s − 30.0·92-s − 14.3·100-s + 31.0·106-s − 3.06·109-s + 9.03·113-s + 4·121-s + 0.0887·127-s + 303.·128-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(623.2600131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(623.2600131\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 \) |
| good | 11 | \( ( 1 - 2 p T^{2} + 243 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 6 T^{2} - 133 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 50 T^{2} + 1227 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 64 T^{2} + 2226 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 80 T^{2} + 3042 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 90 T^{2} + 4283 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 118 T^{2} + 6363 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 120 T^{2} + 6818 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2 T^{2} + 3339 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 5 T + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 100 T^{2} + 7542 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 36 T^{2} + 1622 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 232 T^{2} + 23058 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 136 T^{2} + 10962 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12 T + 164 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 36 T^{2} - 3178 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 220 T^{2} + 26022 T^{4} + 220 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 232 T^{2} + 27954 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.97584453811611342084176372054, −3.97087576304114748576120426539, −3.86493439617128081875218355729, −3.75025368342775154685767485459, −3.72101810907084682765045100331, −3.30252083323582653005041356019, −3.27310248044070726681009933952, −3.24931348693851286001597524205, −3.10368446984060542256961683050, −3.00623969996457179039722433107, −3.00454868268232223648477811474, −2.64471678903771777019667879169, −2.53070511784759051123165775547, −2.41731649365692021992534308305, −2.40291947256406299287977371099, −2.05435750262812371804935934256, −2.03661580200571410342539942931, −1.95451636120438803249937965831, −1.90771236697076819342610197999, −1.72008004637556065979444017874, −1.46847700278065354273376826997, −0.889456567586859239066329556391, −0.77169847761621113723630462291, −0.67456089818154571045477312015, −0.50866062040120126303082961106,
0.50866062040120126303082961106, 0.67456089818154571045477312015, 0.77169847761621113723630462291, 0.889456567586859239066329556391, 1.46847700278065354273376826997, 1.72008004637556065979444017874, 1.90771236697076819342610197999, 1.95451636120438803249937965831, 2.03661580200571410342539942931, 2.05435750262812371804935934256, 2.40291947256406299287977371099, 2.41731649365692021992534308305, 2.53070511784759051123165775547, 2.64471678903771777019667879169, 3.00454868268232223648477811474, 3.00623969996457179039722433107, 3.10368446984060542256961683050, 3.24931348693851286001597524205, 3.27310248044070726681009933952, 3.30252083323582653005041356019, 3.72101810907084682765045100331, 3.75025368342775154685767485459, 3.86493439617128081875218355729, 3.97087576304114748576120426539, 3.97584453811611342084176372054
Plot not available for L-functions of degree greater than 10.
