| L(s) = 1 | − 8·2-s + 20·4-s + 24·8-s − 4·9-s − 246·16-s + 32·18-s − 4·23-s − 12·25-s + 408·32-s − 80·36-s + 32·46-s + 18·49-s + 96·50-s + 756·64-s − 56·71-s − 96·72-s + 10·81-s − 80·92-s − 144·98-s − 240·100-s + 48·121-s + 127-s − 3.96e3·128-s + 131-s + 137-s + 139-s + 448·142-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 10·4-s + 8.48·8-s − 4/3·9-s − 61.5·16-s + 7.54·18-s − 0.834·23-s − 2.39·25-s + 72.1·32-s − 13.3·36-s + 4.71·46-s + 18/7·49-s + 13.5·50-s + 94.5·64-s − 6.64·71-s − 11.3·72-s + 10/9·81-s − 8.34·92-s − 14.5·98-s − 24·100-s + 4.36·121-s + 0.0887·127-s − 350.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 37.5·142-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.008969819971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008969819971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 - 18 T^{2} + 162 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T + p T^{2} )^{8} \) |
| 5 | \( ( 1 + 6 T^{2} + 42 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 24 T^{2} + 318 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 16 T^{2} + 334 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 54 T^{2} + 1290 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 14 T^{2} - 62 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 20 T^{2} + 934 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 108 T^{2} + 5382 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 32 T^{2} + 286 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 162 T^{2} + 10242 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 136 T^{2} + 8430 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 86 T^{2} + 6090 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 184 T^{2} + 14814 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 204 T^{2} + 17574 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 18 T^{2} - 1566 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 148 T^{2} + 15046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 50 T^{2} - 3230 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 12 T^{2} - 3594 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 230 T^{2} + 27690 T^{4} + 230 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 348 T^{2} + 48822 T^{4} + 348 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.66941846206314280002021966940, −4.52930538343297573116387095020, −4.50869192671804496693723338230, −4.48747997145691799631919707719, −4.44026794814075506803798950562, −4.17963334364974551901377929254, −4.03998191228707960984719580580, −3.95376045125836643136137813582, −3.71726077528271441226837759712, −3.58898018818012977223070877253, −3.48446786288200031030421076833, −3.18171637714577314613753289873, −3.08026622977014306256533459985, −2.76797445958528355038052902729, −2.23721860614929952723521194738, −2.20316464611195749207424933439, −1.95657745549044615126283028970, −1.76108333271764270090120167372, −1.62097130163417770685436426656, −1.43257673872000474012704787056, −1.20468534085558514213513833396, −0.898173715919261999745550697673, −0.52909655831044588668860575424, −0.45753798898387082510068007954, −0.23153414582622329670873325908,
0.23153414582622329670873325908, 0.45753798898387082510068007954, 0.52909655831044588668860575424, 0.898173715919261999745550697673, 1.20468534085558514213513833396, 1.43257673872000474012704787056, 1.62097130163417770685436426656, 1.76108333271764270090120167372, 1.95657745549044615126283028970, 2.20316464611195749207424933439, 2.23721860614929952723521194738, 2.76797445958528355038052902729, 3.08026622977014306256533459985, 3.18171637714577314613753289873, 3.48446786288200031030421076833, 3.58898018818012977223070877253, 3.71726077528271441226837759712, 3.95376045125836643136137813582, 4.03998191228707960984719580580, 4.17963334364974551901377929254, 4.44026794814075506803798950562, 4.48747997145691799631919707719, 4.50869192671804496693723338230, 4.52930538343297573116387095020, 4.66941846206314280002021966940
Plot not available for L-functions of degree greater than 10.
