| L(s) = 1 | + 2-s + 3-s − 2·5-s + 6-s − 3·7-s − 8-s − 2·10-s + 2·11-s − 2·13-s − 3·14-s − 2·15-s − 16-s − 2·17-s − 3·21-s + 2·22-s + 3·23-s − 24-s + 5·25-s − 2·26-s − 27-s − 4·29-s − 2·30-s − 8·31-s + 2·33-s − 2·34-s + 6·35-s + 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.894·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 0.632·10-s + 0.603·11-s − 0.554·13-s − 0.801·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.654·21-s + 0.426·22-s + 0.625·23-s − 0.204·24-s + 25-s − 0.392·26-s − 0.192·27-s − 0.742·29-s − 0.365·30-s − 1.43·31-s + 0.348·33-s − 0.342·34-s + 1.01·35-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.750317331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750317331\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 79 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−11.37142667067888359319492808577, −10.73075190981091123825852340143, −10.60316854183755170660479097049, −9.595485974127469630764471351761, −9.409175889405804377230841325103, −9.056079510638091044522256165459, −8.756637278987005886543599569253, −7.75688017036650433856386732303, −7.74738883131058030124798879803, −7.03751154020237226182898587718, −6.73954299629789577432883693199, −5.87570665004468509850137549841, −5.81679689936112792883309162455, −4.81607956045615971838870492692, −4.39982971774830111623549309510, −3.81761906656828092493663204851, −3.46662019962385906445248482861, −2.79206264506806920546610692371, −2.23297910760392907017667257840, −0.70147731866239897110176586516,
0.70147731866239897110176586516, 2.23297910760392907017667257840, 2.79206264506806920546610692371, 3.46662019962385906445248482861, 3.81761906656828092493663204851, 4.39982971774830111623549309510, 4.81607956045615971838870492692, 5.81679689936112792883309162455, 5.87570665004468509850137549841, 6.73954299629789577432883693199, 7.03751154020237226182898587718, 7.74738883131058030124798879803, 7.75688017036650433856386732303, 8.756637278987005886543599569253, 9.056079510638091044522256165459, 9.409175889405804377230841325103, 9.595485974127469630764471351761, 10.60316854183755170660479097049, 10.73075190981091123825852340143, 11.37142667067888359319492808577
