| L(s) = 1 | + 2·3-s − 2·4-s − 2·5-s + 3·9-s − 4·11-s − 4·12-s + 2·13-s − 4·15-s − 10·19-s + 4·20-s − 2·23-s − 7·25-s + 4·27-s − 6·29-s + 2·31-s − 8·33-s − 6·36-s − 8·37-s + 4·39-s + 16·41-s + 2·43-s + 8·44-s − 6·45-s + 2·47-s − 4·52-s + 2·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 0.894·5-s + 9-s − 1.20·11-s − 1.15·12-s + 0.554·13-s − 1.03·15-s − 2.29·19-s + 0.894·20-s − 0.417·23-s − 7/5·25-s + 0.769·27-s − 1.11·29-s + 0.359·31-s − 1.39·33-s − 36-s − 1.31·37-s + 0.640·39-s + 2.49·41-s + 0.304·43-s + 1.20·44-s − 0.894·45-s + 0.291·47-s − 0.554·52-s + 0.274·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 105 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 168 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 193 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 135 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 145 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.802788799349662912668121023762, −8.760251703819577128001627280330, −8.074066595591203925246228321664, −8.070618728059663424381349750674, −7.66473641656575889689639890996, −7.25436371632256678799256700772, −6.78806105220811687161784593936, −6.19412075463920484268310558303, −5.62103796878376426609962436957, −5.55394744710465133331229211670, −4.53570714635406993888613817034, −4.35935818842015462933099405280, −4.05169929909269610530377443167, −3.81284491185912862588283291525, −3.01667820728312016077012342912, −2.61014193119911064757557484744, −2.06783125106069819011629902465, −1.44882733476222978687682703199, 0, 0,
1.44882733476222978687682703199, 2.06783125106069819011629902465, 2.61014193119911064757557484744, 3.01667820728312016077012342912, 3.81284491185912862588283291525, 4.05169929909269610530377443167, 4.35935818842015462933099405280, 4.53570714635406993888613817034, 5.55394744710465133331229211670, 5.62103796878376426609962436957, 6.19412075463920484268310558303, 6.78806105220811687161784593936, 7.25436371632256678799256700772, 7.66473641656575889689639890996, 8.070618728059663424381349750674, 8.074066595591203925246228321664, 8.760251703819577128001627280330, 8.802788799349662912668121023762
