| L(s) = 1 | − 64·11-s + 20·13-s − 1.92e3·23-s − 3.35e3·25-s + 2.97e3·37-s − 3.62e4·47-s − 2.20e4·49-s − 8.07e4·59-s + 2.51e4·61-s − 1.41e5·71-s − 5.24e4·73-s − 1.99e5·83-s − 1.73e5·97-s − 2.81e5·107-s + 2.31e5·109-s − 3.19e5·121-s + 127-s + 131-s + 137-s + 139-s − 1.28e3·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.42e5·169-s + ⋯ |
| L(s) = 1 | − 0.159·11-s + 0.0328·13-s − 0.756·23-s − 1.07·25-s + 0.356·37-s − 2.39·47-s − 1.31·49-s − 3.02·59-s + 0.865·61-s − 3.33·71-s − 1.15·73-s − 3.18·83-s − 1.87·97-s − 2.37·107-s + 1.86·109-s − 1.98·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 0.00523·143-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 1.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3354 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 22030 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 32 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1901410 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 654458 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 960 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 4043274 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47998 p^{2} T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 1486 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 173317458 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 163859062 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 18112 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 810254586 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 40384 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12582 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2544375910 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 70912 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 26202 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 654944734 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 99936 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4580977806 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 86930 T + p^{5} T^{2} )^{2} \) |
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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
−10.66697026366531657324553215227, −10.26010139076654564805456472799, −9.696966426173627737915910066820, −9.518410866698042672893391126988, −8.738744093914801271577076364564, −8.345033494341877753038566652131, −7.74739473759362906963349647183, −7.51833715920358056763826555391, −6.68348054262413984642519264923, −6.29275387286691101221956677173, −5.72539386225555045093745774557, −5.23462765085304508197602979914, −4.32112173969150695947156195769, −4.24133335183177943831953266376, −3.10781577645451579971166659651, −2.89748612553254709020585396226, −1.68729024722548789957196405662, −1.50415806897680816549869756930, 0, 0,
1.50415806897680816549869756930, 1.68729024722548789957196405662, 2.89748612553254709020585396226, 3.10781577645451579971166659651, 4.24133335183177943831953266376, 4.32112173969150695947156195769, 5.23462765085304508197602979914, 5.72539386225555045093745774557, 6.29275387286691101221956677173, 6.68348054262413984642519264923, 7.51833715920358056763826555391, 7.74739473759362906963349647183, 8.345033494341877753038566652131, 8.738744093914801271577076364564, 9.518410866698042672893391126988, 9.696966426173627737915910066820, 10.26010139076654564805456472799, 10.66697026366531657324553215227
