| L(s) = 1 | − 3-s − 2·4-s − 2·5-s − 3·7-s − 9-s − 2·11-s + 2·12-s − 6·13-s + 2·15-s − 5·17-s − 5·19-s + 4·20-s + 3·21-s − 4·23-s + 25-s + 6·28-s + 10·29-s − 4·31-s + 2·33-s + 6·35-s + 2·36-s − 3·37-s + 6·39-s + 10·41-s − 7·43-s + 4·44-s + 2·45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s − 1.13·7-s − 1/3·9-s − 0.603·11-s + 0.577·12-s − 1.66·13-s + 0.516·15-s − 1.21·17-s − 1.14·19-s + 0.894·20-s + 0.654·21-s − 0.834·23-s + 1/5·25-s + 1.13·28-s + 1.85·29-s − 0.718·31-s + 0.348·33-s + 1.01·35-s + 1/3·36-s − 0.493·37-s + 0.960·39-s + 1.56·41-s − 1.06·43-s + 0.603·44-s + 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80923 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80923 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 80923 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 389 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 52 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 88 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T - 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 120 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 3 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 4 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 115 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 100 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−14.7688699332, −14.1669927794, −13.7930364332, −13.3166040610, −12.8353291299, −12.4434209798, −12.2565586148, −11.5986400916, −11.2142269133, −10.5308419532, −10.2924324856, −9.65324522479, −9.29341982378, −8.76336974158, −8.22580113403, −7.85707731414, −7.03630710506, −6.73266713116, −6.14598300948, −5.41869716895, −4.88085901351, −4.29128446677, −3.98907020253, −2.90179304629, −2.35819634467, 0, 0,
2.35819634467, 2.90179304629, 3.98907020253, 4.29128446677, 4.88085901351, 5.41869716895, 6.14598300948, 6.73266713116, 7.03630710506, 7.85707731414, 8.22580113403, 8.76336974158, 9.29341982378, 9.65324522479, 10.2924324856, 10.5308419532, 11.2142269133, 11.5986400916, 12.2565586148, 12.4434209798, 12.8353291299, 13.3166040610, 13.7930364332, 14.1669927794, 14.7688699332
