| L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 3·9-s − 7·11-s + 2·13-s − 4·15-s − 17-s − 2·19-s + 2·21-s − 7·23-s + 3·25-s − 4·27-s − 4·29-s − 6·31-s + 14·33-s − 2·35-s − 37-s − 4·39-s − 7·41-s − 4·43-s + 6·45-s + 6·47-s − 9·49-s + 2·51-s + 3·53-s − 14·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 9-s − 2.11·11-s + 0.554·13-s − 1.03·15-s − 0.242·17-s − 0.458·19-s + 0.436·21-s − 1.45·23-s + 3/5·25-s − 0.769·27-s − 0.742·29-s − 1.07·31-s + 2.43·33-s − 0.338·35-s − 0.164·37-s − 0.640·39-s − 1.09·41-s − 0.609·43-s + 0.894·45-s + 0.875·47-s − 9/7·49-s + 0.280·51-s + 0.412·53-s − 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 212 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 144 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−9.129052017223740455350397482064, −9.127824150215530130059355599076, −8.284619757135976812825184874847, −8.090679296162782050925222620871, −7.54114582175150962645937058974, −7.27963111380182083595123314937, −6.48541215611456982530282531878, −6.47555709008817504279660443924, −5.82421214765066569221039474000, −5.66229097552477811332424770824, −5.06930188695849626713046925189, −5.01821916877390181451004967746, −4.14079700723787643925114025994, −3.87483087660444351327862411011, −2.95554076281844851911141349426, −2.69756382682205190569741995420, −1.74425175867601671629138296755, −1.61645514247363549630957226429, 0, 0,
1.61645514247363549630957226429, 1.74425175867601671629138296755, 2.69756382682205190569741995420, 2.95554076281844851911141349426, 3.87483087660444351327862411011, 4.14079700723787643925114025994, 5.01821916877390181451004967746, 5.06930188695849626713046925189, 5.66229097552477811332424770824, 5.82421214765066569221039474000, 6.47555709008817504279660443924, 6.48541215611456982530282531878, 7.27963111380182083595123314937, 7.54114582175150962645937058974, 8.090679296162782050925222620871, 8.284619757135976812825184874847, 9.127824150215530130059355599076, 9.129052017223740455350397482064
