| L(s) = 1 | − 7-s − 3·9-s − 3·11-s − 9·13-s + 3·17-s − 8·19-s − 15·23-s + 10·25-s − 12·29-s + 15·31-s − 12·41-s + 43-s + 7·49-s + 9·53-s − 24·59-s − 4·61-s + 3·63-s − 15·67-s + 15·71-s − 11·73-s + 3·77-s + 27·79-s + 9·81-s + 9·83-s − 3·89-s + 9·91-s + 15·97-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.904·11-s − 2.49·13-s + 0.727·17-s − 1.83·19-s − 3.12·23-s + 2·25-s − 2.22·29-s + 2.69·31-s − 1.87·41-s + 0.152·43-s + 49-s + 1.23·53-s − 3.12·59-s − 0.512·61-s + 0.377·63-s − 1.83·67-s + 1.78·71-s − 1.28·73-s + 0.341·77-s + 3.03·79-s + 81-s + 0.987·83-s − 0.317·89-s + 0.943·91-s + 1.52·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3272956991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3272956991\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T + 98 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−10.62136932479033688976220223273, −10.15959109586737380288293841490, −10.05578221720810885794461222185, −9.422458898376722910259850537903, −9.040280677806997098891337582381, −8.409369503282265000940508556192, −8.126661541163361842691947123854, −7.55585219094294198340138136111, −7.48485912882999473434793616263, −6.45079004648456930263483490906, −6.41449105322008510581625137571, −5.75159481964818817157012895290, −5.28111464735773451035085010996, −4.62974481913457865988849216938, −4.49529691293554234074978131418, −3.51450206437649034852446966750, −2.99299126269127378945328090403, −2.23368192668174533960188615727, −2.15252140079644199989262637486, −0.27694553280123372469941108467,
0.27694553280123372469941108467, 2.15252140079644199989262637486, 2.23368192668174533960188615727, 2.99299126269127378945328090403, 3.51450206437649034852446966750, 4.49529691293554234074978131418, 4.62974481913457865988849216938, 5.28111464735773451035085010996, 5.75159481964818817157012895290, 6.41449105322008510581625137571, 6.45079004648456930263483490906, 7.48485912882999473434793616263, 7.55585219094294198340138136111, 8.126661541163361842691947123854, 8.409369503282265000940508556192, 9.040280677806997098891337582381, 9.422458898376722910259850537903, 10.05578221720810885794461222185, 10.15959109586737380288293841490, 10.62136932479033688976220223273
