| L(s) = 1 | + 8·2-s + 48·4-s + 64·5-s + 256·8-s − 80·9-s + 512·10-s + 1.28e3·16-s − 640·18-s + 3.07e3·20-s + 1.82e3·25-s + 6.14e3·32-s − 3.84e3·36-s + 2.56e3·37-s + 1.63e4·40-s + 3.36e3·41-s − 5.12e3·45-s − 4.72e3·49-s + 1.45e4·50-s − 1.16e4·61-s + 2.86e4·64-s − 2.04e4·72-s − 1.32e4·73-s + 2.04e4·74-s + 8.19e4·80-s − 161·81-s + 2.68e4·82-s − 4.09e4·90-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 2.55·5-s + 4·8-s − 0.987·9-s + 5.11·10-s + 5·16-s − 1.97·18-s + 7.67·20-s + 2.91·25-s + 6·32-s − 2.96·36-s + 1.86·37-s + 10.2·40-s + 2·41-s − 2.52·45-s − 1.96·49-s + 5.83·50-s − 3.14·61-s + 7·64-s − 3.95·72-s − 2.49·73-s + 3.73·74-s + 64/5·80-s − 0.0245·81-s + 4·82-s − 5.05·90-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(17.96964864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.96964864\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 41 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 80 T^{2} + p^{8} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 32 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4720 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 29200 T^{2} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 251120 T^{2} + p^{8} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1280 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7633840 T^{2} + p^{8} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5842 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 35111600 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 17131120 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6640 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 56238800 T^{2} + p^{8} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−12.68413067831482121093803486602, −12.10612200649246152335749212467, −11.38914495857444130338094128359, −11.17852646550035893364238243855, −10.43633026786740657263163278664, −10.16930577055119329471228804205, −9.418745676803683096264091898687, −9.117890117984751309993460028943, −7.972241069899757134591139501261, −7.57262262095874506928610799409, −6.53785511580411871885464232040, −6.27849961600666210927405258628, −5.70450768450917937616401012633, −5.66431257566040090195207335246, −4.78537288682379299808747161972, −4.23065059115940057743388505873, −2.92709703542015435037571378368, −2.78675143180918292755627661725, −1.94290654468113830074086246306, −1.34833985553508993175891843916,
1.34833985553508993175891843916, 1.94290654468113830074086246306, 2.78675143180918292755627661725, 2.92709703542015435037571378368, 4.23065059115940057743388505873, 4.78537288682379299808747161972, 5.66431257566040090195207335246, 5.70450768450917937616401012633, 6.27849961600666210927405258628, 6.53785511580411871885464232040, 7.57262262095874506928610799409, 7.972241069899757134591139501261, 9.117890117984751309993460028943, 9.418745676803683096264091898687, 10.16930577055119329471228804205, 10.43633026786740657263163278664, 11.17852646550035893364238243855, 11.38914495857444130338094128359, 12.10612200649246152335749212467, 12.68413067831482121093803486602
