L(s) = 1 | − 3-s + 4-s + 9-s − 12-s − 6·13-s + 16-s + 4·19-s − 2·25-s − 27-s + 36-s − 4·37-s + 6·39-s + 13·43-s − 48-s − 14·49-s − 6·52-s − 4·57-s − 22·61-s + 64-s + 4·67-s + 8·73-s + 2·75-s + 4·76-s + 81-s − 11·97-s − 2·100-s − 12·103-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.66·13-s + 1/4·16-s + 0.917·19-s − 2/5·25-s − 0.192·27-s + 1/6·36-s − 0.657·37-s + 0.960·39-s + 1.98·43-s − 0.144·48-s − 2·49-s − 0.832·52-s − 0.529·57-s − 2.81·61-s + 1/8·64-s + 0.488·67-s + 0.936·73-s + 0.230·75-s + 0.458·76-s + 1/9·81-s − 1.11·97-s − 1/5·100-s − 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 12 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−8.138832278295180988992082641359, −7.79380793231723333815239822508, −7.33232044179123697279254900974, −7.14118420978245559693474438413, −6.39540539772989034460122647747, −6.14504142692034810129969354879, −5.45233820964288072326165090096, −5.09762044448004847739559428121, −4.63290052093302179442069274802, −4.02929678652284225029587421276, −3.27110713302371014006833787942, −2.73680352755498545831319167822, −2.07121897929449582590926347310, −1.26307831324452862795634492913, 0,
1.26307831324452862795634492913, 2.07121897929449582590926347310, 2.73680352755498545831319167822, 3.27110713302371014006833787942, 4.02929678652284225029587421276, 4.63290052093302179442069274802, 5.09762044448004847739559428121, 5.45233820964288072326165090096, 6.14504142692034810129969354879, 6.39540539772989034460122647747, 7.14118420978245559693474438413, 7.33232044179123697279254900974, 7.79380793231723333815239822508, 8.138832278295180988992082641359