| L(s) = 1 | + 8.14e3·16-s + 2.30e4·31-s − 1.30e6·61-s − 7.08e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 1.98·16-s + 0.773·31-s − 5.74·61-s − 4·121-s + 2.95·256-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.011171152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.011171152\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 8143 T^{4} + p^{24} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 432524144062082 T^{4} + p^{24} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 79567922 T^{2} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 37214344554868798 T^{4} + p^{24} T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5758 T + p^{6} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 51418335280297668478 T^{4} + p^{24} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + \)\(34\!\cdots\!22\)\( T^{4} + p^{24} T^{8} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 325798 T + p^{6} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 312456859202 T^{2} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - \)\(25\!\cdots\!38\)\( T^{4} + p^{24} T^{8} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
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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
−7.74723427967648781477635894766, −7.69414122151757266122485188085, −7.50069244098975670841353578162, −6.96109978385052592400057026056, −6.71490177952829739985922791343, −6.42219207520296614567700317944, −6.15830001684040812990378932291, −5.97745320185312081084554623510, −5.63760131461261305131407252462, −5.32097239550951793097313932182, −5.14066180745811438672235683359, −4.70265992994240745823852580546, −4.31517964439098245471140816798, −4.31263262712049890022579057967, −3.74125807096154175082939186822, −3.42493668544256926033175152335, −3.07441924960011540198534837648, −2.99473664360891983423847914091, −2.51850685861921317458136825981, −2.12444991108389098217413050397, −1.51593121538760239691075256467, −1.37457156125979319537285306224, −1.16246512836799807988093919421, −0.59806899849924898431532497661, −0.13202966658187260725483337767,
0.13202966658187260725483337767, 0.59806899849924898431532497661, 1.16246512836799807988093919421, 1.37457156125979319537285306224, 1.51593121538760239691075256467, 2.12444991108389098217413050397, 2.51850685861921317458136825981, 2.99473664360891983423847914091, 3.07441924960011540198534837648, 3.42493668544256926033175152335, 3.74125807096154175082939186822, 4.31263262712049890022579057967, 4.31517964439098245471140816798, 4.70265992994240745823852580546, 5.14066180745811438672235683359, 5.32097239550951793097313932182, 5.63760131461261305131407252462, 5.97745320185312081084554623510, 6.15830001684040812990378932291, 6.42219207520296614567700317944, 6.71490177952829739985922791343, 6.96109978385052592400057026056, 7.50069244098975670841353578162, 7.69414122151757266122485188085, 7.74723427967648781477635894766
