| L(s) = 1 | − 3-s + 2·5-s − 4·11-s + 12·13-s − 2·15-s − 2·17-s − 4·19-s + 8·23-s + 5·25-s + 27-s − 4·29-s + 4·33-s + 10·37-s − 12·39-s − 12·41-s + 8·43-s + 2·51-s − 6·53-s − 8·55-s + 4·57-s + 4·59-s − 6·61-s + 24·65-s + 4·67-s − 8·69-s − 16·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.20·11-s + 3.32·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 25-s + 0.192·27-s − 0.742·29-s + 0.696·33-s + 1.64·37-s − 1.92·39-s − 1.87·41-s + 1.21·43-s + 0.280·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + 2.97·65-s + 0.488·67-s − 0.963·69-s − 1.89·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.398069873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.398069873\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.368475444754522374718561456056, −8.660913775422952447732167881809, −8.590540788984670057295091452881, −8.153532025306567319842610407143, −7.67891367640837341167791734360, −7.12732579585037915469358335368, −6.74825564270252250658741036469, −6.23089322094063686615257245394, −6.12167169919231988386940114456, −5.78831819601328764971785715378, −5.32903068299519592921866715257, −4.87543922481801154558251501036, −4.48827772370948461906822379338, −3.84002387962593328066017770461, −3.53308399213546321049714109111, −2.75673251700723453675903034931, −2.64478692983707578553363107074, −1.54300822451132350383270739731, −1.44392184869960712883829266041, −0.57515332802897908637646260890,
0.57515332802897908637646260890, 1.44392184869960712883829266041, 1.54300822451132350383270739731, 2.64478692983707578553363107074, 2.75673251700723453675903034931, 3.53308399213546321049714109111, 3.84002387962593328066017770461, 4.48827772370948461906822379338, 4.87543922481801154558251501036, 5.32903068299519592921866715257, 5.78831819601328764971785715378, 6.12167169919231988386940114456, 6.23089322094063686615257245394, 6.74825564270252250658741036469, 7.12732579585037915469358335368, 7.67891367640837341167791734360, 8.153532025306567319842610407143, 8.590540788984670057295091452881, 8.660913775422952447732167881809, 9.368475444754522374718561456056
