| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 3·5-s + 9·6-s − 7·7-s − 3·8-s + 3·9-s + 9·10-s − 12·12-s − 2·13-s + 21·14-s + 9·15-s + 3·16-s − 4·17-s − 9·18-s + 2·19-s − 12·20-s + 21·21-s − 5·23-s + 9·24-s + 2·25-s + 6·26-s − 28·28-s − 27·30-s − 6·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s − 2.64·7-s − 1.06·8-s + 9-s + 2.84·10-s − 3.46·12-s − 0.554·13-s + 5.61·14-s + 2.32·15-s + 3/4·16-s − 0.970·17-s − 2.12·18-s + 0.458·19-s − 2.68·20-s + 4.58·21-s − 1.04·23-s + 1.83·24-s + 2/5·25-s + 1.17·26-s − 5.29·28-s − 4.92·30-s − 1.07·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5209 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5209 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 142 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 103 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 18 T + 152 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 89 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 135 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−18.1125918254, −17.3457581075, −16.9017465444, −16.6173844317, −16.3018366308, −15.8355023907, −15.4246197882, −14.7817826375, −13.4896847959, −13.0663594060, −12.3412246867, −11.9904777029, −11.4729345107, −10.9792288003, −10.1069773200, −9.98623587000, −9.39610745538, −8.78924366219, −8.03852771648, −7.34890569539, −6.71314927907, −6.23209022904, −5.48593877076, −4.12855127510, −3.14754990651, 0, 0,
3.14754990651, 4.12855127510, 5.48593877076, 6.23209022904, 6.71314927907, 7.34890569539, 8.03852771648, 8.78924366219, 9.39610745538, 9.98623587000, 10.1069773200, 10.9792288003, 11.4729345107, 11.9904777029, 12.3412246867, 13.0663594060, 13.4896847959, 14.7817826375, 15.4246197882, 15.8355023907, 16.3018366308, 16.6173844317, 16.9017465444, 17.3457581075, 18.1125918254
