| L(s) = 1 | − 2-s + 2·4-s − 3·7-s − 5·8-s − 2·11-s − 2·13-s + 3·14-s + 5·16-s + 8·17-s − 16·19-s + 2·22-s − 3·23-s + 2·26-s − 6·28-s − 29-s − 10·32-s − 8·34-s + 8·37-s + 16·38-s + 5·41-s − 8·43-s − 4·44-s + 3·46-s − 7·47-s + 7·49-s − 4·52-s − 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 1.13·7-s − 1.76·8-s − 0.603·11-s − 0.554·13-s + 0.801·14-s + 5/4·16-s + 1.94·17-s − 3.67·19-s + 0.426·22-s − 0.625·23-s + 0.392·26-s − 1.13·28-s − 0.185·29-s − 1.76·32-s − 1.37·34-s + 1.31·37-s + 2.59·38-s + 0.780·41-s − 1.21·43-s − 0.603·44-s + 0.442·46-s − 1.02·47-s + 49-s − 0.554·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3820116304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3820116304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.91405946265766374870851749679, −10.25808217177034498632273759870, −9.895183945586868705403190120068, −9.401494894108609174218915590874, −9.133939079565944715854203308098, −8.470043208626085190084086385835, −8.226927594636734235714688426153, −7.66024324288023754575593224679, −7.33722671160166898074880901519, −6.44951178544787877921830913337, −6.44108223885072474479195228379, −5.99612353950498737137371204888, −5.62169403098116426301867502272, −4.68635528239629710217673204112, −4.20189868866273516923037271542, −3.22906221062972436043939803167, −3.15278084172945382871504386796, −2.32785361445130168953501196763, −1.85580140037294899991126995904, −0.34066087748587836479031662230,
0.34066087748587836479031662230, 1.85580140037294899991126995904, 2.32785361445130168953501196763, 3.15278084172945382871504386796, 3.22906221062972436043939803167, 4.20189868866273516923037271542, 4.68635528239629710217673204112, 5.62169403098116426301867502272, 5.99612353950498737137371204888, 6.44108223885072474479195228379, 6.44951178544787877921830913337, 7.33722671160166898074880901519, 7.66024324288023754575593224679, 8.226927594636734235714688426153, 8.470043208626085190084086385835, 9.133939079565944715854203308098, 9.401494894108609174218915590874, 9.895183945586868705403190120068, 10.25808217177034498632273759870, 10.91405946265766374870851749679
