| L(s) = 1 | + 12·2-s + 70·4-s + 4·7-s + 264·8-s + 6·9-s + 48·14-s + 729·16-s + 72·18-s − 4·19-s + 10·25-s + 280·28-s + 1.60e3·32-s + 420·36-s − 48·38-s − 12·41-s − 24·47-s + 34·49-s + 120·50-s + 1.05e3·56-s − 12·59-s + 24·63-s + 3.06e3·64-s − 24·67-s − 12·71-s + 1.58e3·72-s − 280·76-s + 22·81-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 35·4-s + 1.51·7-s + 93.3·8-s + 2·9-s + 12.8·14-s + 182.·16-s + 16.9·18-s − 0.917·19-s + 2·25-s + 52.9·28-s + 284.·32-s + 70·36-s − 7.78·38-s − 1.87·41-s − 3.50·47-s + 34/7·49-s + 16.9·50-s + 141.·56-s − 1.56·59-s + 3.02·63-s + 383.·64-s − 2.93·67-s − 1.42·71-s + 186.·72-s − 32.1·76-s + 22/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(872.2199682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(872.2199682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 3 | \( 1 - 2 p T^{2} + 14 T^{4} - 8 p T^{6} + 79 T^{8} - 8 p^{3} T^{10} + 14 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 - 5 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 + 2 T^{2} + 70 T^{4} - 808 T^{6} - 24881 T^{8} - 808 p^{2} T^{10} + 70 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 54 T^{2} + 1654 T^{4} - 36936 T^{6} + 666399 T^{8} - 36936 p^{2} T^{10} + 1654 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 7 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 + 38 T^{2} + 1014 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 102 T^{2} + 4238 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 56 T^{2} + 1767 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T - 35 T^{2} - 66 T^{3} + 1884 T^{4} - 66 p T^{5} - 35 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 78 T^{2} + 3070 T^{4} + 53352 T^{6} - 5583921 T^{8} + 53352 p^{2} T^{10} + 3070 p^{4} T^{12} - 78 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 + 4 T^{2} + 874 T^{4} - 25904 T^{6} - 7199261 T^{8} - 25904 p^{2} T^{10} + 874 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 6 T - 11 T^{2} - 426 T^{3} - 3396 T^{4} - 426 p T^{5} - 11 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 30 T^{2} + 62 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 6 T - 95 T^{2} - 66 T^{3} + 9564 T^{4} - 66 p T^{5} - 95 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 128 T^{2} + 11055 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 190 T^{2} + 18238 T^{4} - 1022200 T^{6} + 54800863 T^{8} - 1022200 p^{2} T^{10} + 18238 p^{4} T^{12} - 190 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 156 T^{2} + 17447 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 86 T^{2} + 7566 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 7 T + p T^{2} )^{8} \) |
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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
−4.45616205812294243222859517793, −4.29239752928085699878730340213, −4.18573898230919691209262755907, −3.92017055504253646069073535816, −3.88430349662771912351977390043, −3.79629688923818590413286925721, −3.76236406528942971547124724460, −3.63720423363257536236207110281, −3.60031220386006664996698464801, −3.10340142104226912934817451413, −3.09020429810435355713124070890, −3.04956746895836336875379014082, −3.01279340836675752018292135730, −2.89043964156180262728687126617, −2.66471075011528838648739515231, −2.36369550693743985738672673935, −2.35094707746500028249797768808, −2.21264936014318618258699664624, −1.65059763698703105866690815670, −1.59923253466223188326047614869, −1.49297488491669257150666899869, −1.48445986689059585449791403409, −1.30001055532120730369052752256, −0.993279212698727776238547259557, −0.32377480651846747146300828799,
0.32377480651846747146300828799, 0.993279212698727776238547259557, 1.30001055532120730369052752256, 1.48445986689059585449791403409, 1.49297488491669257150666899869, 1.59923253466223188326047614869, 1.65059763698703105866690815670, 2.21264936014318618258699664624, 2.35094707746500028249797768808, 2.36369550693743985738672673935, 2.66471075011528838648739515231, 2.89043964156180262728687126617, 3.01279340836675752018292135730, 3.04956746895836336875379014082, 3.09020429810435355713124070890, 3.10340142104226912934817451413, 3.60031220386006664996698464801, 3.63720423363257536236207110281, 3.76236406528942971547124724460, 3.79629688923818590413286925721, 3.88430349662771912351977390043, 3.92017055504253646069073535816, 4.18573898230919691209262755907, 4.29239752928085699878730340213, 4.45616205812294243222859517793
Plot not available for L-functions of degree greater than 10.
