| L(s) = 1 | + 16·13-s − 16·25-s + 16·37-s − 24·49-s + 48·61-s + 16·73-s − 32·97-s + 48·109-s − 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4.43·13-s − 3.19·25-s + 2.63·37-s − 3.42·49-s + 6.14·61-s + 1.87·73-s − 3.24·97-s + 4.59·109-s − 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(28.16175868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.16175868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 8 T^{2} + 58 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 12 T^{2} + 18 p T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 28 T^{2} + 406 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 28 T^{2} + 486 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 12 T^{2} - 42 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 60 T^{2} + 1830 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 40 T^{2} + 2010 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 12 T^{2} + 1566 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 76 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 60 T^{2} + 1670 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 44 T^{2} + 310 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 28 T^{2} - 1658 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 136 T^{2} + 10170 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 108 T^{2} + 7830 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 156 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 140 T^{2} + 13366 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 28 T^{2} + 2086 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 300 T^{2} + 34974 T^{4} + 300 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 60 T^{2} + 13110 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 164 T^{2} + 20518 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.28578962553955428312505753458, −2.82287743139708732535461160820, −2.82075625826789992970348907517, −2.79882165478119030720936055212, −2.71422810300987160597209991638, −2.57029507668472210559366400074, −2.54704021010595873182273207404, −2.42205827811976498670287354402, −2.39658923507398812647961456220, −1.89024472101790649096190756793, −1.83950456717572386003103173045, −1.79014630503868381663221616460, −1.76585111144122028862934721576, −1.73579672227852271443544395487, −1.70262166382499381677067255032, −1.54938191010047960103194307268, −1.43078898818115153972548286113, −1.05473686253184175075013371543, −0.958584870362284638526655673468, −0.837882700675547215529375618224, −0.805331971034362009150194999818, −0.63988251630590270384826828515, −0.56940833519122272556297177446, −0.39344218012415796689052972061, −0.19779492308396867995413960966,
0.19779492308396867995413960966, 0.39344218012415796689052972061, 0.56940833519122272556297177446, 0.63988251630590270384826828515, 0.805331971034362009150194999818, 0.837882700675547215529375618224, 0.958584870362284638526655673468, 1.05473686253184175075013371543, 1.43078898818115153972548286113, 1.54938191010047960103194307268, 1.70262166382499381677067255032, 1.73579672227852271443544395487, 1.76585111144122028862934721576, 1.79014630503868381663221616460, 1.83950456717572386003103173045, 1.89024472101790649096190756793, 2.39658923507398812647961456220, 2.42205827811976498670287354402, 2.54704021010595873182273207404, 2.57029507668472210559366400074, 2.71422810300987160597209991638, 2.79882165478119030720936055212, 2.82075625826789992970348907517, 2.82287743139708732535461160820, 3.28578962553955428312505753458
Plot not available for L-functions of degree greater than 10.
