| L(s) = 1 | + 84·5-s − 27·9-s + 364·13-s − 492·17-s + 4.04e3·25-s − 156·29-s − 1.06e3·37-s − 1.83e3·41-s − 2.26e3·45-s − 1.00e3·49-s + 9.25e3·53-s − 2.69e3·61-s + 3.05e4·65-s − 1.85e3·73-s + 729·81-s − 4.13e4·85-s + 2.31e4·89-s − 2.62e4·97-s + 1.09e4·101-s + 3.23e4·109-s + 3.68e3·113-s − 9.82e3·117-s + 2.88e4·121-s + 1.38e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3.35·5-s − 1/3·9-s + 2.15·13-s − 1.70·17-s + 6.46·25-s − 0.185·29-s − 0.774·37-s − 1.09·41-s − 1.11·45-s − 0.418·49-s + 3.29·53-s − 0.723·61-s + 7.23·65-s − 0.347·73-s + 1/9·81-s − 5.72·85-s + 2.92·89-s − 2.78·97-s + 1.07·101-s + 2.72·109-s + 0.288·113-s − 0.717·117-s + 1.97·121-s + 8.88·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(6.033301001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.033301001\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 42 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 1006 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 28850 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 14 p T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 246 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 246770 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 190 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 78 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 330670 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 530 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 918 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6111410 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4550206 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4626 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 24182450 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1346 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 39119090 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 47477954 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 926 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 58545362 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 48908686 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11586 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13118 T + p^{4} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19555631875313904409188874271, −11.44869314188011670224108434895, −10.90523813072270801231993214845, −10.58544026464937081782700756380, −10.09606577228384718873383102606, −9.726960182644347710969659176516, −8.975716046987329542309371260056, −8.687702816114288192447939600830, −8.660633982726283366491846776974, −7.27533644260360429654371665584, −6.44921764496777643039157612605, −6.43757257973687223372246432545, −5.83282924104122986019724831100, −5.43583839228069757058869336980, −4.80960687606415568008633795096, −3.78998382436171677216514922442, −2.87635204302111670242438518160, −2.04714792667969420759364680045, −1.80115705674542605631928302001, −0.913792393046645537420334612113,
0.913792393046645537420334612113, 1.80115705674542605631928302001, 2.04714792667969420759364680045, 2.87635204302111670242438518160, 3.78998382436171677216514922442, 4.80960687606415568008633795096, 5.43583839228069757058869336980, 5.83282924104122986019724831100, 6.43757257973687223372246432545, 6.44921764496777643039157612605, 7.27533644260360429654371665584, 8.660633982726283366491846776974, 8.687702816114288192447939600830, 8.975716046987329542309371260056, 9.726960182644347710969659176516, 10.09606577228384718873383102606, 10.58544026464937081782700756380, 10.90523813072270801231993214845, 11.44869314188011670224108434895, 12.19555631875313904409188874271
