| L(s) = 1 | + 2·2-s + 2·3-s − 5-s + 4·6-s − 2·7-s − 4·8-s − 2·9-s − 2·10-s + 5·11-s − 7·13-s − 4·14-s − 2·15-s − 4·16-s + 6·17-s − 4·18-s − 2·19-s − 4·21-s + 10·22-s − 23-s − 8·24-s − 8·25-s − 14·26-s − 10·27-s + 29-s − 4·30-s − 6·31-s + 10·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 0.447·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s − 2/3·9-s − 0.632·10-s + 1.50·11-s − 1.94·13-s − 1.06·14-s − 0.516·15-s − 16-s + 1.45·17-s − 0.942·18-s − 0.458·19-s − 0.872·21-s + 2.13·22-s − 0.208·23-s − 1.63·24-s − 8/5·25-s − 2.74·26-s − 1.92·27-s + 0.185·29-s − 0.730·30-s − 1.07·31-s + 1.74·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126859 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126859 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 126859 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 551 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 75 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 144 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−14.1365271269, −13.7892985069, −13.3849115706, −12.6873882708, −12.5178523601, −12.0304607532, −11.7792938769, −11.3470478460, −10.4464028039, −9.83882389366, −9.52310361336, −9.21113374780, −8.74508401508, −8.22996773976, −7.56546799788, −7.36737609255, −6.50804317311, −5.82812832506, −5.62618745069, −4.89296998750, −4.22462916366, −3.77856062767, −3.37445133131, −2.86162534874, −2.02804104924, 0,
2.02804104924, 2.86162534874, 3.37445133131, 3.77856062767, 4.22462916366, 4.89296998750, 5.62618745069, 5.82812832506, 6.50804317311, 7.36737609255, 7.56546799788, 8.22996773976, 8.74508401508, 9.21113374780, 9.52310361336, 9.83882389366, 10.4464028039, 11.3470478460, 11.7792938769, 12.0304607532, 12.5178523601, 12.6873882708, 13.3849115706, 13.7892985069, 14.1365271269
