| L(s) = 1 | + 20·5-s − 28·13-s − 28·17-s + 250·25-s − 28·37-s − 200·41-s + 68·53-s − 120·61-s − 560·65-s − 28·73-s + 18·81-s − 560·85-s − 172·97-s + 408·101-s − 332·113-s − 364·121-s + 2.50e3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 392·169-s + ⋯ |
| L(s) = 1 | + 4·5-s − 2.15·13-s − 1.64·17-s + 10·25-s − 0.756·37-s − 4.87·41-s + 1.28·53-s − 1.96·61-s − 8.61·65-s − 0.383·73-s + 2/9·81-s − 6.58·85-s − 1.77·97-s + 4.03·101-s − 2.93·113-s − 3.00·121-s + 20·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.31·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.965212534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.965212534\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 2 p^{2} T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 178 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 182 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2}( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 61262 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1282 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1018 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 1853102 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 1068658 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )^{2}( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 25029778 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 5222 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{2}( 1 + 110 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 11522 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 74812142 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 14242 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 86 T + 3698 T^{2} + 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.488325864564110064581683256990, −8.062921832567536196073228203383, −7.48782368186654049772429293738, −7.46197642840250498827044796478, −6.91940553677748806329272712866, −6.84232245743665062645500094227, −6.58423626327924228612723213300, −6.48164058502388670360661652085, −6.27542103870411346160943133753, −5.79100704168248318847113850030, −5.54069434275770773466755425310, −5.20740305592310443953479439652, −5.10766120381115257430719227950, −5.05001222096255797289958278746, −4.54278384763710180356947004381, −4.41776675821689257190447966640, −3.69726692994974759735688889635, −3.18592380176611164734463271359, −2.85638087870136002659864614092, −2.66648193813883301703559552269, −2.19582484051172517402059104390, −1.98172176408769609344098022438, −1.65537253415396284051143400428, −1.42292377574634100821604015922, −0.37551774289463986235792382995,
0.37551774289463986235792382995, 1.42292377574634100821604015922, 1.65537253415396284051143400428, 1.98172176408769609344098022438, 2.19582484051172517402059104390, 2.66648193813883301703559552269, 2.85638087870136002659864614092, 3.18592380176611164734463271359, 3.69726692994974759735688889635, 4.41776675821689257190447966640, 4.54278384763710180356947004381, 5.05001222096255797289958278746, 5.10766120381115257430719227950, 5.20740305592310443953479439652, 5.54069434275770773466755425310, 5.79100704168248318847113850030, 6.27542103870411346160943133753, 6.48164058502388670360661652085, 6.58423626327924228612723213300, 6.84232245743665062645500094227, 6.91940553677748806329272712866, 7.46197642840250498827044796478, 7.48782368186654049772429293738, 8.062921832567536196073228203383, 8.488325864564110064581683256990
