| L(s) = 1 | − 2·2-s − 3-s − 4-s + 2·6-s + 8·8-s + 9-s + 12-s − 7·16-s − 2·18-s − 19-s − 8·24-s − 6·25-s − 27-s − 4·29-s − 14·32-s − 36-s + 2·38-s + 4·41-s − 8·43-s + 7·48-s − 14·49-s + 12·50-s + 12·53-s + 2·54-s + 57-s + 8·58-s + 24·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 1/2·4-s + 0.816·6-s + 2.82·8-s + 1/3·9-s + 0.288·12-s − 7/4·16-s − 0.471·18-s − 0.229·19-s − 1.63·24-s − 6/5·25-s − 0.192·27-s − 0.742·29-s − 2.47·32-s − 1/6·36-s + 0.324·38-s + 0.624·41-s − 1.21·43-s + 1.01·48-s − 2·49-s + 1.69·50-s + 1.64·53-s + 0.272·54-s + 0.132·57-s + 1.05·58-s + 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2996296504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2996296504\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.211204361001959275319801150204, −8.700339636508216566718463872719, −8.204224733564106943471891272665, −7.962026255736907116683171399764, −7.43259308735181576062150293394, −6.88281746236137995334549719629, −6.37820814688737049661034336254, −5.42974656732833448500123753190, −5.30285359031178015236684552035, −4.62864849822679992384179631378, −3.91841567788800315953565193679, −3.69904169102384607719527879903, −2.25179019930198933577153857080, −1.48488050469889056862198933364, −0.49711700208122723907813480784,
0.49711700208122723907813480784, 1.48488050469889056862198933364, 2.25179019930198933577153857080, 3.69904169102384607719527879903, 3.91841567788800315953565193679, 4.62864849822679992384179631378, 5.30285359031178015236684552035, 5.42974656732833448500123753190, 6.37820814688737049661034336254, 6.88281746236137995334549719629, 7.43259308735181576062150293394, 7.962026255736907116683171399764, 8.204224733564106943471891272665, 8.700339636508216566718463872719, 9.211204361001959275319801150204
