| L(s) = 1 | + 4-s + 4·7-s − 3·16-s + 2·19-s − 25-s + 4·28-s + 8·31-s + 8·37-s − 4·43-s + 2·49-s − 12·61-s − 7·64-s − 8·67-s + 12·73-s + 2·76-s + 8·79-s + 8·97-s − 100-s + 24·103-s + 12·109-s − 12·112-s − 2·121-s + 8·124-s + 127-s + 131-s + 8·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s − 3/4·16-s + 0.458·19-s − 1/5·25-s + 0.755·28-s + 1.43·31-s + 1.31·37-s − 0.609·43-s + 2/7·49-s − 1.53·61-s − 7/8·64-s − 0.977·67-s + 1.40·73-s + 0.229·76-s + 0.900·79-s + 0.812·97-s − 0.0999·100-s + 2.36·103-s + 1.14·109-s − 1.13·112-s − 0.181·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.821651881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821651881\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.167798775468386922497017119471, −7.81629925534961684383397522072, −7.56286056811260075809566840048, −7.01862500787748876169081978041, −6.39492674973921678255633336802, −6.19358130523662731614596938467, −5.54933623644349374683330696508, −4.92586372856273525185184956084, −4.62565435014953298320952525244, −4.29428437177532962119033259859, −3.40322077346350739176953316718, −2.88611679888280351701569505957, −2.15652647337383028968047721099, −1.72158885252222435895990718582, −0.853110982786161162683313101739,
0.853110982786161162683313101739, 1.72158885252222435895990718582, 2.15652647337383028968047721099, 2.88611679888280351701569505957, 3.40322077346350739176953316718, 4.29428437177532962119033259859, 4.62565435014953298320952525244, 4.92586372856273525185184956084, 5.54933623644349374683330696508, 6.19358130523662731614596938467, 6.39492674973921678255633336802, 7.01862500787748876169081978041, 7.56286056811260075809566840048, 7.81629925534961684383397522072, 8.167798775468386922497017119471
