| L(s) = 1 | + 138·9-s + 568·11-s − 2.99e3·23-s − 634·25-s − 8.73e3·29-s − 2.52e4·37-s + 2.71e3·43-s + 2.83e4·53-s + 7.28e3·67-s − 7.12e4·71-s + 1.09e5·79-s − 4.00e4·81-s + 7.83e4·99-s − 4.36e5·107-s − 1.92e5·109-s − 2.74e5·113-s − 8.01e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.67e5·169-s + ⋯ |
| L(s) = 1 | + 0.567·9-s + 1.41·11-s − 1.17·23-s − 0.202·25-s − 1.92·29-s − 3.03·37-s + 0.223·43-s + 1.38·53-s + 0.198·67-s − 1.67·71-s + 1.96·79-s − 0.677·81-s + 0.803·99-s − 3.68·107-s − 1.54·109-s − 2.02·113-s − 0.497·121-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 − 1.25·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 46 p T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 634 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 284 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 35954 p T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2817250 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 229142 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 1496 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4366 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 15722366 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 12630 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 142552786 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1356 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 357849118 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14150 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 31458982 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 422082602 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 3644 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 35632 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2484166610 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 54616 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7877806102 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 10752624754 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16535359486 T^{2} + p^{10} T^{4} \) |
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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
−9.304157760557911844059016534439, −9.122088764093297733115932505245, −8.392475806858590853188477822154, −8.172718042027476660665226186262, −7.51449968228387528747885239951, −7.07998832666188692121151544876, −6.83004571985001800130970358256, −6.32776804144520293125650234293, −5.75338627191633291496452714150, −5.38396525912450219756291133537, −4.88993830044251946887750886344, −4.06379010492289707084024266804, −3.82367318563608491722258838700, −3.64158735997528612557168836267, −2.67647287694976980505609222745, −2.05836302611049870111941187047, −1.48396571567936548585173378335, −1.23775167043699346356161627177, 0, 0,
1.23775167043699346356161627177, 1.48396571567936548585173378335, 2.05836302611049870111941187047, 2.67647287694976980505609222745, 3.64158735997528612557168836267, 3.82367318563608491722258838700, 4.06379010492289707084024266804, 4.88993830044251946887750886344, 5.38396525912450219756291133537, 5.75338627191633291496452714150, 6.32776804144520293125650234293, 6.83004571985001800130970358256, 7.07998832666188692121151544876, 7.51449968228387528747885239951, 8.172718042027476660665226186262, 8.392475806858590853188477822154, 9.122088764093297733115932505245, 9.304157760557911844059016534439
