| L(s) = 1 | + 2·9-s − 10·11-s − 4·19-s + 6·29-s − 8·31-s − 4·41-s − 49-s − 8·59-s − 12·61-s − 10·71-s − 26·79-s − 5·81-s − 20·99-s − 36·101-s − 10·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 8·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 3.01·11-s − 0.917·19-s + 1.11·29-s − 1.43·31-s − 0.624·41-s − 1/7·49-s − 1.04·59-s − 1.53·61-s − 1.18·71-s − 2.92·79-s − 5/9·81-s − 2.01·99-s − 3.58·101-s − 0.957·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.413086864234618494873052695891, −8.133685211579967905936760095884, −8.064499817756993566215652899860, −7.41331755143394449844783082373, −7.08094170044652930332358686339, −7.01884629898722068219912796360, −6.15274325143092457654797980340, −5.96027751751010371658031391778, −5.34752915786799963321742546700, −5.28379601360898431998292633966, −4.54782058936440789788789985950, −4.49333666902057687959281748280, −3.90668781513470109811689282590, −3.01712644156276181559076837877, −2.99469315841616912960814690385, −2.45718304305413496758433000770, −1.84205090641577382071622873419, −1.33611695416700141873696129895, 0, 0,
1.33611695416700141873696129895, 1.84205090641577382071622873419, 2.45718304305413496758433000770, 2.99469315841616912960814690385, 3.01712644156276181559076837877, 3.90668781513470109811689282590, 4.49333666902057687959281748280, 4.54782058936440789788789985950, 5.28379601360898431998292633966, 5.34752915786799963321742546700, 5.96027751751010371658031391778, 6.15274325143092457654797980340, 7.01884629898722068219912796360, 7.08094170044652930332358686339, 7.41331755143394449844783082373, 8.064499817756993566215652899860, 8.133685211579967905936760095884, 8.413086864234618494873052695891
