| L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·15-s + 2·19-s + 4·23-s − 25-s − 27-s + 2·29-s + 6·43-s − 2·45-s + 2·49-s − 16·53-s − 2·57-s − 2·67-s − 4·69-s − 10·71-s + 22·73-s + 75-s + 81-s − 2·87-s − 4·95-s + 20·97-s − 16·101-s − 8·115-s + 16·121-s + 12·125-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.516·15-s + 0.458·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.914·43-s − 0.298·45-s + 2/7·49-s − 2.19·53-s − 0.264·57-s − 0.244·67-s − 0.481·69-s − 1.18·71-s + 2.57·73-s + 0.115·75-s + 1/9·81-s − 0.214·87-s − 0.410·95-s + 2.03·97-s − 1.59·101-s − 0.746·115-s + 1.45·121-s + 1.07·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9924375519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9924375519\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 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
−9.219294567737737741418191517077, −8.741376433667275230554324050566, −8.070076176227226852141592377786, −7.73969349440985799618779448886, −7.34701321807468149897168674541, −6.70350755558764186505549114258, −6.34535548444757324393800096774, −5.66921546283650403680761952582, −5.15357957517873504176438697726, −4.59868182932519763838184983044, −4.10683540348882130527463000336, −3.41543724697528163110247717742, −2.85134360002445027828003992115, −1.78603492656205691487762164203, −0.69048694540391895169385497150,
0.69048694540391895169385497150, 1.78603492656205691487762164203, 2.85134360002445027828003992115, 3.41543724697528163110247717742, 4.10683540348882130527463000336, 4.59868182932519763838184983044, 5.15357957517873504176438697726, 5.66921546283650403680761952582, 6.34535548444757324393800096774, 6.70350755558764186505549114258, 7.34701321807468149897168674541, 7.73969349440985799618779448886, 8.070076176227226852141592377786, 8.741376433667275230554324050566, 9.219294567737737741418191517077
