| L(s) = 1 | − 3·4-s + 328·13-s + 64·16-s + 40·19-s + 174·25-s − 312·31-s − 372·37-s + 656·43-s − 984·52-s − 1.58e3·61-s − 549·64-s + 88·67-s − 252·73-s − 120·76-s + 1.42e3·79-s + 3.19e3·97-s − 522·100-s + 1.83e3·103-s + 684·109-s + 762·121-s + 936·124-s + 127-s + 131-s + 137-s + 139-s + 1.11e3·148-s + 149-s + ⋯ |
| L(s) = 1 | − 3/8·4-s + 6.99·13-s + 16-s + 0.482·19-s + 1.39·25-s − 1.80·31-s − 1.65·37-s + 2.32·43-s − 2.62·52-s − 3.31·61-s − 1.07·64-s + 0.160·67-s − 0.404·73-s − 0.181·76-s + 2.02·79-s + 3.34·97-s − 0.521·100-s + 1.75·103-s + 0.601·109-s + 0.572·121-s + 0.677·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.619·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.23290990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.23290990\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} - 55 T^{4} + 3 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 174 T^{2} + 14651 T^{4} - 174 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 762 T^{2} - 1190917 T^{4} - 762 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 3670 T^{2} - 10668669 T^{4} - 3670 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 20 T - 6459 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 7234 T^{2} - 95705133 T^{4} - 7234 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10806 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 156 T - 5455 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 186 T - 16057 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 110406 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 164 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 13970 T^{2} - 10584054429 T^{4} + 13970 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 273130 T^{2} + 52435635771 T^{4} - 273130 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 386134 T^{2} + 106918932315 T^{4} - 386134 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 790 T + 397119 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 44 T - 298827 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 518146 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 126 T - 373141 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 712 T + 13905 T^{2} - 712 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 1001450 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 709626 T^{2} + 6587768915 T^{4} + 709626 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 798 T + p^{3} T^{2} )^{4} \) |
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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
−7.55278344955188792904598624353, −7.41022030818004945805399211349, −7.26200212171286659135273105964, −6.59019401071679902935456369969, −6.55820020896357741908156028801, −6.30565039487957488188998368270, −6.01173958692304985015283803229, −5.87969980156041434560871635682, −5.83809284178082110387639760235, −5.32616702437604258227520314024, −5.31058702682546867168624957692, −4.54666221634896053416572156107, −4.53292754729179105222149203450, −4.03496542144304994086587733365, −3.80779913016288870499181617590, −3.62904616598105243269841064289, −3.23683773719698926479807349488, −3.18819520425538686316765557667, −3.14305005065303859085076511527, −1.95142947827024229334032153547, −1.85596846019093968333992202662, −1.37840826886658875057442099131, −1.20311353026688041507781994523, −0.72696858692961876479507106578, −0.66990451374232947092039446573,
0.66990451374232947092039446573, 0.72696858692961876479507106578, 1.20311353026688041507781994523, 1.37840826886658875057442099131, 1.85596846019093968333992202662, 1.95142947827024229334032153547, 3.14305005065303859085076511527, 3.18819520425538686316765557667, 3.23683773719698926479807349488, 3.62904616598105243269841064289, 3.80779913016288870499181617590, 4.03496542144304994086587733365, 4.53292754729179105222149203450, 4.54666221634896053416572156107, 5.31058702682546867168624957692, 5.32616702437604258227520314024, 5.83809284178082110387639760235, 5.87969980156041434560871635682, 6.01173958692304985015283803229, 6.30565039487957488188998368270, 6.55820020896357741908156028801, 6.59019401071679902935456369969, 7.26200212171286659135273105964, 7.41022030818004945805399211349, 7.55278344955188792904598624353
