| L(s) = 1 | + 5·9-s − 8·11-s − 12·19-s − 2·29-s − 18·31-s + 6·41-s + 10·49-s + 8·59-s − 20·61-s − 10·71-s + 12·79-s + 16·81-s + 16·89-s − 40·99-s − 20·101-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 60·171-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 2.41·11-s − 2.75·19-s − 0.371·29-s − 3.23·31-s + 0.937·41-s + 10/7·49-s + 1.04·59-s − 2.56·61-s − 1.18·71-s + 1.35·79-s + 16/9·81-s + 1.69·89-s − 4.02·99-s − 1.99·101-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s − 4.58·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.175229023606042385992057234139, −7.72192308001022868304892860098, −7.35056794366390380867642246525, −7.27122553736599112400227658848, −6.74461630248785559806007287921, −6.38055219821832046866988926071, −5.74983199475273923966557123214, −5.71531811598472010858017923047, −5.11782060213045759144569176827, −4.82381956575589086694887641647, −4.32320847214229393609346927889, −4.07650839955192711372972108393, −3.68150592729800116861449404187, −3.12792862105891749147473144489, −2.42637272773495874480543280662, −2.18597920951845860254381719540, −1.88076293422317677242153232157, −1.15946400338431335773317452442, 0, 0,
1.15946400338431335773317452442, 1.88076293422317677242153232157, 2.18597920951845860254381719540, 2.42637272773495874480543280662, 3.12792862105891749147473144489, 3.68150592729800116861449404187, 4.07650839955192711372972108393, 4.32320847214229393609346927889, 4.82381956575589086694887641647, 5.11782060213045759144569176827, 5.71531811598472010858017923047, 5.74983199475273923966557123214, 6.38055219821832046866988926071, 6.74461630248785559806007287921, 7.27122553736599112400227658848, 7.35056794366390380867642246525, 7.72192308001022868304892860098, 8.175229023606042385992057234139
