| L(s) = 1 | + 2-s − 2·3-s − 2·5-s − 2·6-s − 3·7-s + 8-s − 2·10-s + 11-s − 4·13-s − 3·14-s + 4·15-s − 16-s + 2·17-s − 9·19-s + 6·21-s + 22-s − 2·23-s − 2·24-s + 3·25-s − 4·26-s + 2·27-s + 10·29-s + 4·30-s − 6·32-s − 2·33-s + 2·34-s + 6·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 0.894·5-s − 0.816·6-s − 1.13·7-s + 0.353·8-s − 0.632·10-s + 0.301·11-s − 1.10·13-s − 0.801·14-s + 1.03·15-s − 1/4·16-s + 0.485·17-s − 2.06·19-s + 1.30·21-s + 0.213·22-s − 0.417·23-s − 0.408·24-s + 3/5·25-s − 0.784·26-s + 0.384·27-s + 1.85·29-s + 0.730·30-s − 1.06·32-s − 0.348·33-s + 0.342·34-s + 1.01·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11423 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11423 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11423 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 52 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - T - 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 115 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 75 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 159 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 64 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 235 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−16.4779584529, −16.2273003065, −15.8944653660, −15.1708479922, −14.6034684906, −14.3957060960, −13.5842924900, −13.0829653851, −12.5553356080, −12.2336002829, −11.8347178525, −11.1927906096, −10.7675558501, −10.0792483630, −9.75474043650, −8.62429365065, −8.43954110584, −7.32953694900, −6.88847701358, −6.28728465565, −5.72009023766, −4.82026896723, −4.40797213108, −3.63150303679, −2.55012424115, 0,
2.55012424115, 3.63150303679, 4.40797213108, 4.82026896723, 5.72009023766, 6.28728465565, 6.88847701358, 7.32953694900, 8.43954110584, 8.62429365065, 9.75474043650, 10.0792483630, 10.7675558501, 11.1927906096, 11.8347178525, 12.2336002829, 12.5553356080, 13.0829653851, 13.5842924900, 14.3957060960, 14.6034684906, 15.1708479922, 15.8944653660, 16.2273003065, 16.4779584529
