| L(s) = 1 | + 3·4-s + 9-s − 2·13-s + 5·16-s + 8·17-s − 4·23-s − 2·25-s + 4·29-s + 3·36-s − 16·43-s + 49-s − 6·52-s + 20·53-s + 4·61-s + 3·64-s + 24·68-s − 8·79-s + 81-s − 12·92-s − 6·100-s − 8·101-s − 12·107-s − 20·113-s + 12·116-s − 2·117-s − 10·121-s + 127-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1/3·9-s − 0.554·13-s + 5/4·16-s + 1.94·17-s − 0.834·23-s − 2/5·25-s + 0.742·29-s + 1/2·36-s − 2.43·43-s + 1/7·49-s − 0.832·52-s + 2.74·53-s + 0.512·61-s + 3/8·64-s + 2.91·68-s − 0.900·79-s + 1/9·81-s − 1.25·92-s − 3/5·100-s − 0.796·101-s − 1.16·107-s − 1.88·113-s + 1.11·116-s − 0.184·117-s − 0.909·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.185201482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.185201482\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−10.04060167430973625690266395105, −9.509789235627489810465129961065, −8.651472367673802417867505956894, −8.087122868124748556333819782542, −7.77561300480119923795680807540, −7.11919441702277745134242616849, −6.84791804438814148842718143863, −6.24628037433267946771516149900, −5.55154188786331842470106103432, −5.27814792294585055557768488491, −4.26012648427532404918209751723, −3.55184179503814644720599334297, −2.89512763316926805245058442270, −2.17237344320845611717407226816, −1.31980219556990438978714861945,
1.31980219556990438978714861945, 2.17237344320845611717407226816, 2.89512763316926805245058442270, 3.55184179503814644720599334297, 4.26012648427532404918209751723, 5.27814792294585055557768488491, 5.55154188786331842470106103432, 6.24628037433267946771516149900, 6.84791804438814148842718143863, 7.11919441702277745134242616849, 7.77561300480119923795680807540, 8.087122868124748556333819782542, 8.651472367673802417867505956894, 9.509789235627489810465129961065, 10.04060167430973625690266395105
