| L(s) = 1 | + 4·2-s + 4·4-s − 16·8-s − 20·11-s − 64·16-s − 80·22-s − 80·23-s + 38·25-s − 32·29-s − 64·32-s + 10·37-s − 38·43-s − 80·44-s − 320·46-s + 152·50-s + 64·53-s − 128·58-s + 192·64-s + 118·67-s + 52·71-s + 40·74-s + 94·79-s − 152·86-s + 320·88-s − 320·92-s + 152·100-s + 256·106-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 1.81·11-s − 4·16-s − 3.63·22-s − 3.47·23-s + 1.51·25-s − 1.10·29-s − 2·32-s + 0.270·37-s − 0.883·43-s − 1.81·44-s − 6.95·46-s + 3.03·50-s + 1.20·53-s − 2.20·58-s + 3·64-s + 1.76·67-s + 0.732·71-s + 0.540·74-s + 1.18·79-s − 1.76·86-s + 3.63·88-s − 3.47·92-s + 1.51·100-s + 2.41·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.523288334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.523288334\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 38 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1895 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2774 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 19 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1718 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5234 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7010 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10295 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13190 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1970 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16466 T^{2} + p^{4} T^{4} \) |
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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
−11.49588151210880412045722221856, −10.66799111421722802970615261659, −10.39586958938698253065616868606, −9.656958157981444180553790093198, −9.631135568187401592027393297465, −8.674737151361210410568088984483, −8.452497462825935363703214115977, −7.902208267446285365438033513244, −7.44443732159623405133782881671, −6.51721579752535754627995806548, −6.28261950114234096757431701208, −5.55608024105503326567480082606, −5.40659281292638505493849979010, −4.88270968907515351332821506890, −4.36161344772521350909584893672, −3.72705594348361441509601377793, −3.47422812957797721484404239867, −2.51292706455095562416273649900, −2.23707232647181870059442763616, −0.35291620435379557324672201019,
0.35291620435379557324672201019, 2.23707232647181870059442763616, 2.51292706455095562416273649900, 3.47422812957797721484404239867, 3.72705594348361441509601377793, 4.36161344772521350909584893672, 4.88270968907515351332821506890, 5.40659281292638505493849979010, 5.55608024105503326567480082606, 6.28261950114234096757431701208, 6.51721579752535754627995806548, 7.44443732159623405133782881671, 7.902208267446285365438033513244, 8.452497462825935363703214115977, 8.674737151361210410568088984483, 9.631135568187401592027393297465, 9.656958157981444180553790093198, 10.39586958938698253065616868606, 10.66799111421722802970615261659, 11.49588151210880412045722221856
