| L(s) = 1 | − 2·3-s + 2·4-s + 3·5-s + 3·9-s − 4·12-s − 6·15-s + 6·20-s − 3·23-s + 4·25-s − 4·27-s − 4·31-s + 6·36-s + 2·37-s + 9·45-s − 3·47-s + 4·49-s − 6·53-s + 3·59-s − 12·60-s − 8·64-s − 4·67-s + 6·69-s − 3·71-s − 8·75-s + 5·81-s + 3·89-s − 6·92-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4-s + 1.34·5-s + 9-s − 1.15·12-s − 1.54·15-s + 1.34·20-s − 0.625·23-s + 4/5·25-s − 0.769·27-s − 0.718·31-s + 36-s + 0.328·37-s + 1.34·45-s − 0.437·47-s + 4/7·49-s − 0.824·53-s + 0.390·59-s − 1.54·60-s − 64-s − 0.488·67-s + 0.722·69-s − 0.356·71-s − 0.923·75-s + 5/9·81-s + 0.317·89-s − 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.13769525321656948639436590464, −6.69243350185303817379573537415, −6.44310983299421548882067057503, −6.11340913414956618458754919580, −5.68819495759312838132936180222, −5.38086424547565149198700647252, −4.98032193823229449699335896154, −4.35263962301717241650736623285, −3.97767135717232869460276552512, −3.20564531800342706767460720185, −2.66888506241133297183477036962, −2.10151938021844476995064373803, −1.71624951086665592459460693802, −1.13230946339803826046642567548, 0,
1.13230946339803826046642567548, 1.71624951086665592459460693802, 2.10151938021844476995064373803, 2.66888506241133297183477036962, 3.20564531800342706767460720185, 3.97767135717232869460276552512, 4.35263962301717241650736623285, 4.98032193823229449699335896154, 5.38086424547565149198700647252, 5.68819495759312838132936180222, 6.11340913414956618458754919580, 6.44310983299421548882067057503, 6.69243350185303817379573537415, 7.13769525321656948639436590464
