| L(s) = 1 | + 2-s + 2·5-s − 5·7-s − 8-s + 2·10-s − 11-s − 14·13-s − 5·14-s − 16-s + 2·17-s − 22-s − 8·23-s + 5·25-s − 14·26-s + 10·29-s − 4·31-s + 2·34-s − 10·35-s − 4·37-s − 2·40-s − 8·41-s − 16·43-s − 8·46-s + 2·47-s + 18·49-s + 5·50-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.894·5-s − 1.88·7-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 3.88·13-s − 1.33·14-s − 1/4·16-s + 0.485·17-s − 0.213·22-s − 1.66·23-s + 25-s − 2.74·26-s + 1.85·29-s − 0.718·31-s + 0.342·34-s − 1.69·35-s − 0.657·37-s − 0.316·40-s − 1.24·41-s − 2.43·43-s − 1.17·46-s + 0.291·47-s + 18/7·49-s + 0.707·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.463651737599295965675675668980, −9.347146940890400031167254519453, −8.402711167269090113813159075923, −8.393748599256950342727763959778, −7.38053985947741429936379499473, −7.35142333821652578905366075368, −6.87195576413003465722387076695, −6.35447981703526983183437500642, −6.22151945108275359559471990606, −5.43014031734927876090125655462, −5.15038466116120683249202141835, −4.91386442592581204627499716929, −4.34586621938657870757983730193, −3.68612678543057485420691162952, −3.06702934538345590599517576196, −2.73874047689079527364115532002, −2.41477780911271363056326392541, −1.67638202708494261981625072922, 0, 0,
1.67638202708494261981625072922, 2.41477780911271363056326392541, 2.73874047689079527364115532002, 3.06702934538345590599517576196, 3.68612678543057485420691162952, 4.34586621938657870757983730193, 4.91386442592581204627499716929, 5.15038466116120683249202141835, 5.43014031734927876090125655462, 6.22151945108275359559471990606, 6.35447981703526983183437500642, 6.87195576413003465722387076695, 7.35142333821652578905366075368, 7.38053985947741429936379499473, 8.393748599256950342727763959778, 8.402711167269090113813159075923, 9.347146940890400031167254519453, 9.463651737599295965675675668980
