| L(s) = 1 | − 12·9-s + 20·25-s − 56·49-s + 90·81-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 240·225-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4·9-s + 4·25-s − 8·49-s + 10·81-s + 8/11·121-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 − 4.30·169-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 + 0.0669·223-s − 16·225-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1735499645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1735499645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| good | 7 | \( ( 1 + p T^{2} )^{8} \) |
| 11 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + p T^{2} )^{8} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
| 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
−4.48550660108274917741018141967, −4.36209133669922089419828175344, −3.86153309243992113464743401193, −3.74530959842003235850763108461, −3.60373583683361334787751366159, −3.59401384314558231938282281350, −3.54882392359838091800767831410, −3.30383803349316740622780350511, −3.09095662046688303152071536076, −3.08566290167545235081967977943, −2.90659460804496379766342987903, −2.88248998634005809457063568149, −2.80215764896064679445822353098, −2.63452261318683327804498077503, −2.25414915125911134041742348184, −2.18815287057687500052051311580, −2.15634412165681408961504156973, −1.97299800824828430034409551543, −1.55501838110925425058075047723, −1.29277513293041586433928372928, −1.28065355013966270050395778982, −1.20384066763146820477574700898, −0.59597037120211312759395947692, −0.46866808024713195643204059910, −0.07687759300264864242748861938,
0.07687759300264864242748861938, 0.46866808024713195643204059910, 0.59597037120211312759395947692, 1.20384066763146820477574700898, 1.28065355013966270050395778982, 1.29277513293041586433928372928, 1.55501838110925425058075047723, 1.97299800824828430034409551543, 2.15634412165681408961504156973, 2.18815287057687500052051311580, 2.25414915125911134041742348184, 2.63452261318683327804498077503, 2.80215764896064679445822353098, 2.88248998634005809457063568149, 2.90659460804496379766342987903, 3.08566290167545235081967977943, 3.09095662046688303152071536076, 3.30383803349316740622780350511, 3.54882392359838091800767831410, 3.59401384314558231938282281350, 3.60373583683361334787751366159, 3.74530959842003235850763108461, 3.86153309243992113464743401193, 4.36209133669922089419828175344, 4.48550660108274917741018141967
Plot not available for L-functions of degree greater than 10.
