| L(s) = 1 | − 3-s + 4-s + 2·7-s + 9-s − 12-s + 4·13-s + 16-s − 8·19-s − 2·21-s − 6·25-s − 27-s + 2·28-s − 8·31-s + 36-s − 4·37-s − 4·39-s − 48-s + 3·49-s + 4·52-s + 8·57-s − 28·61-s + 2·63-s + 64-s + 8·67-s + 12·73-s + 6·75-s − 8·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.83·19-s − 0.436·21-s − 6/5·25-s − 0.192·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s − 0.657·37-s − 0.640·39-s − 0.144·48-s + 3/7·49-s + 0.554·52-s + 1.05·57-s − 3.58·61-s + 0.251·63-s + 1/8·64-s + 0.977·67-s + 1.40·73-s + 0.692·75-s − 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.036862420200993690310137538450, −7.82491759382859411774440769361, −7.06103937421558916977777144566, −6.91394332771191655575545668718, −6.17569848665337782312463950999, −5.88974476808873252053152463118, −5.62472658216482534683012385296, −4.72832473817225769292057457440, −4.54468764173297491130716278035, −3.72716170584453900058590617219, −3.52268894932194979183855581758, −2.43951823946059783270435316404, −1.87596574858564202101556695192, −1.37340800303348386328024038090, 0,
1.37340800303348386328024038090, 1.87596574858564202101556695192, 2.43951823946059783270435316404, 3.52268894932194979183855581758, 3.72716170584453900058590617219, 4.54468764173297491130716278035, 4.72832473817225769292057457440, 5.62472658216482534683012385296, 5.88974476808873252053152463118, 6.17569848665337782312463950999, 6.91394332771191655575545668718, 7.06103937421558916977777144566, 7.82491759382859411774440769361, 8.036862420200993690310137538450
