| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 4·5-s + 9·6-s − 4·7-s − 3·8-s + 4·9-s + 12·10-s − 4·11-s − 12·12-s − 13-s + 12·14-s + 12·15-s + 3·16-s − 9·17-s − 12·18-s − 4·19-s − 16·20-s + 12·21-s + 12·22-s + 7·23-s + 9·24-s + 7·25-s + 3·26-s − 16·28-s − 36·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 1.78·5-s + 3.67·6-s − 1.51·7-s − 1.06·8-s + 4/3·9-s + 3.79·10-s − 1.20·11-s − 3.46·12-s − 0.277·13-s + 3.20·14-s + 3.09·15-s + 3/4·16-s − 2.18·17-s − 2.82·18-s − 0.917·19-s − 3.57·20-s + 2.61·21-s + 2.55·22-s + 1.45·23-s + 1.83·24-s + 7/5·25-s + 0.588·26-s − 3.02·28-s − 6.57·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5331 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5331 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 1777 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 67 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 10 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 83 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−18.0090879745, −17.4082405269, −16.9965471829, −16.6067919131, −16.1310316312, −15.6978393930, −15.4037078056, −14.8978999003, −13.4045364698, −12.9176602074, −12.5710112799, −11.8051945184, −11.3695691267, −10.8853467110, −10.4291060753, −10.0272088026, −9.08867449136, −8.70563944438, −8.11718266608, −7.30525665154, −6.80263040825, −6.30873433097, −5.15181542780, −4.35756890285, −3.04371973361, 0, 0,
3.04371973361, 4.35756890285, 5.15181542780, 6.30873433097, 6.80263040825, 7.30525665154, 8.11718266608, 8.70563944438, 9.08867449136, 10.0272088026, 10.4291060753, 10.8853467110, 11.3695691267, 11.8051945184, 12.5710112799, 12.9176602074, 13.4045364698, 14.8978999003, 15.4037078056, 15.6978393930, 16.1310316312, 16.6067919131, 16.9965471829, 17.4082405269, 18.0090879745
