| L(s) = 1 | − 86·3-s − 128·4-s + 5.20e3·9-s + 1.10e4·12-s + 1.63e4·16-s − 1.19e5·19-s − 1.56e5·25-s − 2.59e5·27-s − 6.66e5·36-s − 4.41e5·43-s − 1.40e6·48-s + 1.64e6·49-s + 1.02e7·57-s − 2.09e6·64-s − 7.70e6·67-s − 9.73e6·73-s + 1.34e7·75-s + 1.52e7·76-s + 1.09e7·81-s − 1.98e7·97-s + 2.00e7·100-s + 3.32e7·108-s + 3.87e7·121-s + 127-s + 3.79e7·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.83·3-s − 4-s + 2.38·9-s + 1.83·12-s + 16-s − 3.99·19-s − 2·25-s − 2.54·27-s − 2.38·36-s − 0.845·43-s − 1.83·48-s + 2·49-s + 7.34·57-s − 64-s − 3.12·67-s − 2.92·73-s + 3.67·75-s + 3.99·76-s + 2.29·81-s − 2.21·97-s + 2·100-s + 2.54·108-s + 1.98·121-s + 1.55·129-s + 2.38·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.09043720524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09043720524\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| 3 | $C_2$ | \( 1 + 86 T + p^{7} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 8814 T + p^{7} T^{2} )( 1 + 8814 T + p^{7} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 22182 T + p^{7} T^{2} )( 1 + 22182 T + p^{7} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 59722 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 236886 T + p^{7} T^{2} )( 1 + 236886 T + p^{7} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 220510 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 1030926 T + p^{7} T^{2} )( 1 + 1030926 T + p^{7} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3851302 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4865614 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4808934 T + p^{7} T^{2} )( 1 + 4808934 T + p^{7} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 7073118 T + p^{7} T^{2} )( 1 + 7073118 T + p^{7} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 9938890 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.87935867673897456990769810571, −16.00568279409553866728559496152, −15.00126437757923894232991457727, −14.95695336261605792682930352493, −13.50449432673428973973366087489, −13.24147472871065821570762625647, −12.43298286122196462644410982727, −12.06238031888029804580905839821, −11.14017179216210654386994765246, −10.41583559725682951418451952237, −10.16162142880413169652343819328, −9.004174026651932852428019333412, −8.259122534409409142009744759664, −7.12976400811754221019899135580, −6.09929991137508251499607145987, −5.76646864788073152987435956614, −4.32516755108482547798934031651, −4.30149679988723322061723507578, −1.78063975664933253916535694103, −0.17884534687863084115475567142,
0.17884534687863084115475567142, 1.78063975664933253916535694103, 4.30149679988723322061723507578, 4.32516755108482547798934031651, 5.76646864788073152987435956614, 6.09929991137508251499607145987, 7.12976400811754221019899135580, 8.259122534409409142009744759664, 9.004174026651932852428019333412, 10.16162142880413169652343819328, 10.41583559725682951418451952237, 11.14017179216210654386994765246, 12.06238031888029804580905839821, 12.43298286122196462644410982727, 13.24147472871065821570762625647, 13.50449432673428973973366087489, 14.95695336261605792682930352493, 15.00126437757923894232991457727, 16.00568279409553866728559496152, 16.87935867673897456990769810571
