| L(s) = 1 | + 4-s − 22·7-s − 10·13-s + 16·16-s − 76·19-s − 14·25-s − 22·28-s + 26·31-s + 68·37-s − 58·43-s + 219·49-s − 10·52-s + 44·61-s + 47·64-s − 196·67-s + 152·73-s − 76·76-s − 94·79-s + 220·91-s − 166·97-s − 14·100-s + 308·103-s − 220·109-s − 352·112-s − 98·121-s + 26·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/4·4-s − 3.14·7-s − 0.769·13-s + 16-s − 4·19-s − 0.559·25-s − 0.785·28-s + 0.838·31-s + 1.83·37-s − 1.34·43-s + 4.46·49-s − 0.192·52-s + 0.721·61-s + 0.734·64-s − 2.92·67-s + 2.08·73-s − 76-s − 1.18·79-s + 2.41·91-s − 1.71·97-s − 0.139·100-s + 2.99·103-s − 2.01·109-s − 3.14·112-s − 0.809·121-s + 0.209·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01702858100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01702858100\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 15 T^{4} - p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 39 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )( 1 + 8 T + 39 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 98 T^{2} - 5037 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 5 T - 144 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 254 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 158 T^{2} - 254877 T^{4} + 158 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 622 T^{2} - 320397 T^{4} - 622 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )^{2}( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 2462 T^{2} + 3235683 T^{4} + 2462 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 29 T - 1008 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 3842 T^{2} + 9881283 T^{4} + 3842 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 4322 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 6926 T^{2} + 35852115 T^{4} + 6926 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 22 T - 3237 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 98 T + 5115 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 47 T - 4032 T^{2} + 47 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 9422 T^{2} + 41315763 T^{4} + 9422 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15518 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 83 T - 2520 T^{2} + 83 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.603488407074123444569888604499, −8.537013842339179622220837921870, −8.076496078966937134968597838874, −7.81290201153504987394572065324, −7.43841716307712152971293294235, −7.30258302511833543540288519306, −6.72509125202228570592054511674, −6.70546853297873819569128906798, −6.37687396742554005432793936121, −6.25621466062732754724427572478, −6.03759637955071981351773021023, −5.91097074704277217323049511830, −5.30902110797148014715244464051, −4.98195818144370522020680456737, −4.59828213330791570789425881523, −4.09635734292052543791320448919, −3.96321252921703025727569770627, −3.85355086716239790653999023216, −3.06425374958090914258484904612, −2.87759165453969265393519022291, −2.85094484705109681300083498439, −1.99906680255054633248481885344, −1.99558873786261628415806921071, −0.824222320852339389931391999625, −0.04255362905564600409042443245,
0.04255362905564600409042443245, 0.824222320852339389931391999625, 1.99558873786261628415806921071, 1.99906680255054633248481885344, 2.85094484705109681300083498439, 2.87759165453969265393519022291, 3.06425374958090914258484904612, 3.85355086716239790653999023216, 3.96321252921703025727569770627, 4.09635734292052543791320448919, 4.59828213330791570789425881523, 4.98195818144370522020680456737, 5.30902110797148014715244464051, 5.91097074704277217323049511830, 6.03759637955071981351773021023, 6.25621466062732754724427572478, 6.37687396742554005432793936121, 6.70546853297873819569128906798, 6.72509125202228570592054511674, 7.30258302511833543540288519306, 7.43841716307712152971293294235, 7.81290201153504987394572065324, 8.076496078966937134968597838874, 8.537013842339179622220837921870, 8.603488407074123444569888604499
