| L(s) = 1 | − 3-s − 3·4-s + 4·7-s + 9-s + 3·12-s + 4·13-s + 5·16-s − 4·21-s + 6·25-s − 27-s − 12·28-s − 16·31-s − 3·36-s + 8·37-s − 4·39-s − 5·48-s + 2·49-s − 12·52-s − 12·61-s + 4·63-s − 3·64-s + 4·67-s + 5·73-s − 6·75-s − 12·79-s + 81-s + 12·84-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s + 1.51·7-s + 1/3·9-s + 0.866·12-s + 1.10·13-s + 5/4·16-s − 0.872·21-s + 6/5·25-s − 0.192·27-s − 2.26·28-s − 2.87·31-s − 1/2·36-s + 1.31·37-s − 0.640·39-s − 0.721·48-s + 2/7·49-s − 1.66·52-s − 1.53·61-s + 0.503·63-s − 3/8·64-s + 0.488·67-s + 0.585·73-s − 0.692·75-s − 1.35·79-s + 1/9·81-s + 1.30·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900747 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900747 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| 457 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 26 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.945018114278809811205558519490, −7.66170406073068916340334433189, −7.22464808196773099075262602515, −6.50761632428092117333907091187, −6.09102903000053795512672653221, −5.49852661735854644068749511091, −5.13634081621122251888447317999, −4.89097620767505370416324664659, −4.28118470304150477463851535934, −3.92227552418233884290072679003, −3.44581884404686450790287806483, −2.51850931121389303882919035510, −1.52871890730946335219777266949, −1.18385481320461125472968043380, 0,
1.18385481320461125472968043380, 1.52871890730946335219777266949, 2.51850931121389303882919035510, 3.44581884404686450790287806483, 3.92227552418233884290072679003, 4.28118470304150477463851535934, 4.89097620767505370416324664659, 5.13634081621122251888447317999, 5.49852661735854644068749511091, 6.09102903000053795512672653221, 6.50761632428092117333907091187, 7.22464808196773099075262602515, 7.66170406073068916340334433189, 7.945018114278809811205558519490
