| L(s) = 1 | + 6·9-s − 120·29-s − 216·41-s − 100·49-s − 280·61-s + 27·81-s − 456·89-s − 72·101-s − 136·109-s − 380·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 284·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 4.13·29-s − 5.26·41-s − 2.04·49-s − 4.59·61-s + 1/3·81-s − 5.12·89-s − 0.712·101-s − 1.24·109-s − 3.14·121-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 + 1.68·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08757405120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08757405120\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 542 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 674 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1490 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2690 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5294 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6530 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 4894 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 3170 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 3934 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6674 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13346 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 114 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 17662 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.70919093595165967541596074901, −6.67484956755454685107158347316, −6.55907650273867380013575366662, −6.00590895154301175419360934194, −5.78392039366067350208952422303, −5.61922406967369563162984585860, −5.44133577992963164756793109586, −5.36149102430139969782919438737, −4.78283554237870000153097629124, −4.73120253490252440658528827202, −4.68627805415718248358908095525, −4.09437945903573076789210034301, −4.01987455564630607744948866477, −3.75120483173843694493621255557, −3.48632982265077855393460615826, −3.13567974274412808040457746303, −3.07801167917781216048349993365, −2.80653937374522856400986723019, −2.28954080874580889599029546163, −1.74750140402972816189692388517, −1.63481924997936591081105886660, −1.54680023873501607434056941632, −1.47016003589184991811843303834, −0.38597456602211916837144430186, −0.06396291648917562664956881568,
0.06396291648917562664956881568, 0.38597456602211916837144430186, 1.47016003589184991811843303834, 1.54680023873501607434056941632, 1.63481924997936591081105886660, 1.74750140402972816189692388517, 2.28954080874580889599029546163, 2.80653937374522856400986723019, 3.07801167917781216048349993365, 3.13567974274412808040457746303, 3.48632982265077855393460615826, 3.75120483173843694493621255557, 4.01987455564630607744948866477, 4.09437945903573076789210034301, 4.68627805415718248358908095525, 4.73120253490252440658528827202, 4.78283554237870000153097629124, 5.36149102430139969782919438737, 5.44133577992963164756793109586, 5.61922406967369563162984585860, 5.78392039366067350208952422303, 6.00590895154301175419360934194, 6.55907650273867380013575366662, 6.67484956755454685107158347316, 6.70919093595165967541596074901
