| L(s) = 1 | − 8·13-s + 36·25-s − 16·37-s + 16·49-s − 16·61-s + 32·97-s + 32·109-s − 60·121-s + ⋯ |
| L(s) = 1 | − 2.21·13-s + 36/5·25-s − 2.63·37-s + 16/7·49-s − 2.04·61-s + 3.24·97-s + 3.06·109-s − 5.45·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6582923518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6582923518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 8 T^{2} + 82 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 30 T^{2} + 435 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} + 643 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 2 p T^{2} + 1075 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 68 T^{2} + 2326 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 32 T^{2} + 130 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2 T^{2} - 2909 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 158 T^{2} + 9907 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 96 T^{2} + 4674 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 144 T^{2} + 9650 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12 T^{2} + 4950 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 188 T^{2} + 17302 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 36 T^{2} + 2214 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 68 T^{2} + 11590 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 48 T^{2} + 14226 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 312 T^{2} + 39890 T^{4} - 312 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T + 202 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.81202526339206201072193109714, −3.48905556051278778774434016751, −3.33436628337203070097826466149, −3.32227141642785388427718063770, −3.28067642985945635501159999209, −3.15783682638869167075711512857, −2.91638518486942597311982220628, −2.87942635491509114238144134735, −2.72946666221837666832313152799, −2.68866983188568538875427624617, −2.51496966361520511044804086190, −2.39159547033578138110365018595, −2.32784955211002782325375233801, −2.18112049684008373636152061632, −1.96360063102006756442964558166, −1.78036338241417877725671776498, −1.60755697010537879310378932205, −1.47778048910606094983584748031, −1.35332670831954932469137343308, −1.04569109156607328132697307080, −0.978886286888699154699238429255, −0.888943915657319115161812379590, −0.66977297357075071729858482013, −0.40407955552972155048823204487, −0.06781927387375986283866360860,
0.06781927387375986283866360860, 0.40407955552972155048823204487, 0.66977297357075071729858482013, 0.888943915657319115161812379590, 0.978886286888699154699238429255, 1.04569109156607328132697307080, 1.35332670831954932469137343308, 1.47778048910606094983584748031, 1.60755697010537879310378932205, 1.78036338241417877725671776498, 1.96360063102006756442964558166, 2.18112049684008373636152061632, 2.32784955211002782325375233801, 2.39159547033578138110365018595, 2.51496966361520511044804086190, 2.68866983188568538875427624617, 2.72946666221837666832313152799, 2.87942635491509114238144134735, 2.91638518486942597311982220628, 3.15783682638869167075711512857, 3.28067642985945635501159999209, 3.32227141642785388427718063770, 3.33436628337203070097826466149, 3.48905556051278778774434016751, 3.81202526339206201072193109714
Plot not available for L-functions of degree greater than 10.
