| L(s) = 1 | + 2·3-s − 3·5-s + 2·7-s + 3·9-s − 8·11-s − 3·13-s − 6·15-s + 2·17-s − 8·19-s + 4·21-s + 2·23-s + 5·25-s + 4·27-s + 13·29-s − 2·31-s − 16·33-s − 6·35-s + 11·37-s − 6·39-s + 41-s + 5·43-s − 9·45-s − 15·47-s + 3·49-s + 4·51-s − 2·53-s + 24·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s + 0.755·7-s + 9-s − 2.41·11-s − 0.832·13-s − 1.54·15-s + 0.485·17-s − 1.83·19-s + 0.872·21-s + 0.417·23-s + 25-s + 0.769·27-s + 2.41·29-s − 0.359·31-s − 2.78·33-s − 1.01·35-s + 1.80·37-s − 0.960·39-s + 0.156·41-s + 0.762·43-s − 1.34·45-s − 2.18·47-s + 3/7·49-s + 0.560·51-s − 0.274·53-s + 3.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 13 T + 92 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 96 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 190 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 228 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64971709939500364914916256556, −7.61281398134608590732092216128, −7.30305426305832870301351715178, −6.79838029264534286585467225560, −6.22024323730082386104415434298, −6.21101894566425267893421697640, −5.32637321486472345150759045410, −5.12342173864766875080406065540, −4.72339418726495577628536825309, −4.54168289208179052538929075147, −4.02809502772808730327913048948, −3.94119327514708601803431821617, −2.92626084659497970937035985799, −2.87981364504074882703478227046, −2.73644350892143027180639791080, −2.26307948419910956403881145132, −1.54920649876814086868171167078, −1.08709967068228190391902878698, 0, 0,
1.08709967068228190391902878698, 1.54920649876814086868171167078, 2.26307948419910956403881145132, 2.73644350892143027180639791080, 2.87981364504074882703478227046, 2.92626084659497970937035985799, 3.94119327514708601803431821617, 4.02809502772808730327913048948, 4.54168289208179052538929075147, 4.72339418726495577628536825309, 5.12342173864766875080406065540, 5.32637321486472345150759045410, 6.21101894566425267893421697640, 6.22024323730082386104415434298, 6.79838029264534286585467225560, 7.30305426305832870301351715178, 7.61281398134608590732092216128, 7.64971709939500364914916256556
