| L(s) = 1 | + 4·3-s − 2·5-s + 8·9-s + 4·11-s − 6·13-s − 8·15-s − 4·19-s + 2·25-s + 12·27-s + 6·29-s + 16·31-s + 16·33-s + 2·37-s − 24·39-s − 4·43-s − 16·45-s + 16·47-s − 2·49-s + 2·53-s − 8·55-s − 16·57-s + 12·59-s − 6·61-s + 12·65-s − 4·67-s + 8·75-s + 23·81-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 8/3·9-s + 1.20·11-s − 1.66·13-s − 2.06·15-s − 0.917·19-s + 2/5·25-s + 2.30·27-s + 1.11·29-s + 2.87·31-s + 2.78·33-s + 0.328·37-s − 3.84·39-s − 0.609·43-s − 2.38·45-s + 2.33·47-s − 2/7·49-s + 0.274·53-s − 1.07·55-s − 2.11·57-s + 1.56·59-s − 0.768·61-s + 1.48·65-s − 0.488·67-s + 0.923·75-s + 23/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.450498151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.450498151\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−11.03506102078398959415191232120, −10.46582720824680621975821340463, −10.10662443683471923793578704840, −9.585770316199146499807467568905, −9.317208974386718494987251222433, −8.643084682641881530707278093552, −8.579281457309852325511144098014, −8.032219862017146177475715167061, −7.76733652253301107098772646183, −7.21195266822028914438476455736, −6.59417153694718370966246885287, −6.49186940732716210834184777345, −5.34642786189251725336081061049, −4.50465825769089052782427535100, −4.36116716080932502653133703863, −3.80789483671956572852543918251, −3.08975650707919337857830423684, −2.60152567375513468867481988589, −2.28322453373648801816088451916, −1.03964368906994799498656426612,
1.03964368906994799498656426612, 2.28322453373648801816088451916, 2.60152567375513468867481988589, 3.08975650707919337857830423684, 3.80789483671956572852543918251, 4.36116716080932502653133703863, 4.50465825769089052782427535100, 5.34642786189251725336081061049, 6.49186940732716210834184777345, 6.59417153694718370966246885287, 7.21195266822028914438476455736, 7.76733652253301107098772646183, 8.032219862017146177475715167061, 8.579281457309852325511144098014, 8.643084682641881530707278093552, 9.317208974386718494987251222433, 9.585770316199146499807467568905, 10.10662443683471923793578704840, 10.46582720824680621975821340463, 11.03506102078398959415191232120
