| L(s) = 1 | − 2-s − 4-s + 3·8-s − 9-s − 16-s − 2·17-s + 18-s + 8·19-s − 10·25-s − 5·32-s + 2·34-s + 36-s − 8·38-s − 8·43-s + 2·49-s + 10·50-s − 24·59-s + 7·64-s − 24·67-s + 2·68-s − 3·72-s − 8·76-s + 81-s + 24·83-s + 8·86-s + 20·89-s − 2·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1/3·9-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s − 2·25-s − 0.883·32-s + 0.342·34-s + 1/6·36-s − 1.29·38-s − 1.21·43-s + 2/7·49-s + 1.41·50-s − 3.12·59-s + 7/8·64-s − 2.93·67-s + 0.242·68-s − 0.353·72-s − 0.917·76-s + 1/9·81-s + 2.63·83-s + 0.862·86-s + 2.11·89-s − 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.090594032537686732103395089135, −8.615451952434614667930414636577, −7.900860653729670862169263895362, −7.60595935169870401648096170117, −7.44623840170883548365244775304, −6.47273159107566705483229806147, −6.05490907628685400491216241873, −5.44881183007587606152730843705, −4.87444452782409767096094076150, −4.44310182082410096214841664802, −3.59546300652664291181585701831, −3.19158750626093917850586098771, −2.09925088994484722217684574212, −1.32032952054096010596372886629, 0,
1.32032952054096010596372886629, 2.09925088994484722217684574212, 3.19158750626093917850586098771, 3.59546300652664291181585701831, 4.44310182082410096214841664802, 4.87444452782409767096094076150, 5.44881183007587606152730843705, 6.05490907628685400491216241873, 6.47273159107566705483229806147, 7.44623840170883548365244775304, 7.60595935169870401648096170117, 7.900860653729670862169263895362, 8.615451952434614667930414636577, 9.090594032537686732103395089135
