| L(s) = 1 | + 64·4-s + 3.07e3·16-s + 3.42e3·19-s + 5.44e3·31-s + 3.29e4·49-s + 1.13e5·61-s + 1.31e5·64-s + 2.19e5·76-s + 2.01e5·79-s − 2.66e5·109-s − 3.22e5·121-s + 3.48e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.41e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s + 2.17·19-s + 1.01·31-s + 1.96·49-s + 3.91·61-s + 4·64-s + 4.34·76-s + 3.62·79-s − 2.15·109-s − 2·121-s + 2.03·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-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 + 0.382·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.276068682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.276068682\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 32989 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 141961 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 1711 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2723 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 135214586 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 211108739 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 56927 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2698325411 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4144040686 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 100564 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14411495111 T^{2} + p^{10} T^{4} \) |
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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
−11.65807709002491445058933728566, −11.27275064373820535899655918326, −10.53388525137663684489134688356, −10.47579246510730645472078592014, −9.637708569281348109308726435558, −9.482852481736386759860752571637, −8.400496092143266394377790076234, −8.092792667407989877706673524119, −7.35508522095215685753854071448, −7.22906019182699561313754904596, −6.55609931903575997889018931677, −6.15057811390926630581463099327, −5.28653744020645626499604979846, −5.24725017484214509276782126193, −3.81524134887619496324232675041, −3.47440328665539204454983796001, −2.47850559886211993069216951009, −2.41475497133123208653350340928, −1.20688678904304418736697766709, −0.895406923448672246553255425026,
0.895406923448672246553255425026, 1.20688678904304418736697766709, 2.41475497133123208653350340928, 2.47850559886211993069216951009, 3.47440328665539204454983796001, 3.81524134887619496324232675041, 5.24725017484214509276782126193, 5.28653744020645626499604979846, 6.15057811390926630581463099327, 6.55609931903575997889018931677, 7.22906019182699561313754904596, 7.35508522095215685753854071448, 8.092792667407989877706673524119, 8.400496092143266394377790076234, 9.482852481736386759860752571637, 9.637708569281348109308726435558, 10.47579246510730645472078592014, 10.53388525137663684489134688356, 11.27275064373820535899655918326, 11.65807709002491445058933728566
