| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s − 7-s + 4·9-s + 6·10-s + 2·11-s − 3·12-s − 2·13-s + 2·14-s + 9·15-s + 16-s − 3·17-s − 8·18-s + 19-s − 3·20-s + 3·21-s − 4·22-s − 4·23-s + 25-s + 4·26-s − 28-s − 6·29-s − 18·30-s − 31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 0.377·7-s + 4/3·9-s + 1.89·10-s + 0.603·11-s − 0.866·12-s − 0.554·13-s + 0.534·14-s + 2.32·15-s + 1/4·16-s − 0.727·17-s − 1.88·18-s + 0.229·19-s − 0.670·20-s + 0.654·21-s − 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 1.11·29-s − 3.28·30-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 359 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 114 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 174 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−19.9792106538, −19.3768544424, −18.9087295598, −18.3724983167, −17.8207373097, −17.4724129722, −16.9514753465, −16.4066464564, −16.0299464838, −15.3482491336, −14.7611109666, −13.7536323190, −12.8078703900, −12.0522309383, −11.8676959823, −11.1352738890, −10.7006019347, −9.70008827668, −9.33802546280, −8.32296493864, −7.71918412015, −6.80397864306, −6.08591447925, −4.96180986609, −3.87679222455, 0,
3.87679222455, 4.96180986609, 6.08591447925, 6.80397864306, 7.71918412015, 8.32296493864, 9.33802546280, 9.70008827668, 10.7006019347, 11.1352738890, 11.8676959823, 12.0522309383, 12.8078703900, 13.7536323190, 14.7611109666, 15.3482491336, 16.0299464838, 16.4066464564, 16.9514753465, 17.4724129722, 17.8207373097, 18.3724983167, 18.9087295598, 19.3768544424, 19.9792106538
