| L(s) = 1 | − 2·2-s + 2·4-s + 9-s − 4·16-s + 2·17-s − 2·18-s − 6·25-s + 8·32-s − 4·34-s + 2·36-s − 24·47-s − 10·49-s + 12·50-s − 8·64-s + 4·68-s + 81-s − 20·89-s + 48·94-s + 20·98-s − 12·100-s + 12·103-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 4·144-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1/3·9-s − 16-s + 0.485·17-s − 0.471·18-s − 6/5·25-s + 1.41·32-s − 0.685·34-s + 1/3·36-s − 3.50·47-s − 1.42·49-s + 1.69·50-s − 64-s + 0.485·68-s + 1/9·81-s − 2.11·89-s + 4.95·94-s + 2.02·98-s − 6/5·100-s + 1.18·103-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/3·144-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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.065222949815774938910669430004, −8.377043825991258905935442762904, −8.113978657058836509344424940894, −7.74254342107732758471772335201, −7.22000947833638768610402834358, −6.58980704853126351607343995411, −6.33458324936254626734123217739, −5.50218710956319622544336182536, −4.92903095151455640052641442356, −4.35260176613519713006555508155, −3.61147274420103775188740230272, −2.91897557245007132060009978873, −1.92661203808846315515150827352, −1.38334250010017986650689702950, 0,
1.38334250010017986650689702950, 1.92661203808846315515150827352, 2.91897557245007132060009978873, 3.61147274420103775188740230272, 4.35260176613519713006555508155, 4.92903095151455640052641442356, 5.50218710956319622544336182536, 6.33458324936254626734123217739, 6.58980704853126351607343995411, 7.22000947833638768610402834358, 7.74254342107732758471772335201, 8.113978657058836509344424940894, 8.377043825991258905935442762904, 9.065222949815774938910669430004
