| L(s) = 1 | − 2·2-s − 3-s − 2·5-s + 2·6-s − 4·7-s + 4·8-s − 2·9-s + 4·10-s + 11-s − 13-s + 8·14-s + 2·15-s − 4·16-s + 4·18-s + 19-s + 4·21-s − 2·22-s − 23-s − 4·24-s + 2·26-s + 2·27-s − 6·29-s − 4·30-s + 3·31-s − 33-s + 8·35-s − 11·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 0.894·5-s + 0.816·6-s − 1.51·7-s + 1.41·8-s − 2/3·9-s + 1.26·10-s + 0.301·11-s − 0.277·13-s + 2.13·14-s + 0.516·15-s − 16-s + 0.942·18-s + 0.229·19-s + 0.872·21-s − 0.426·22-s − 0.208·23-s − 0.816·24-s + 0.392·26-s + 0.384·27-s − 1.11·29-s − 0.730·30-s + 0.538·31-s − 0.174·33-s + 1.35·35-s − 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1813 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1813 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 120 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 114 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T - 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 216 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−19.3037507485, −18.7836615640, −18.2052809452, −17.7117129448, −17.0312131871, −16.8581567364, −16.3258496789, −15.6372193758, −15.2511430739, −14.2355808567, −13.7169857556, −13.0679309715, −12.2570319287, −11.9501410659, −11.0639650707, −10.4722640146, −9.83202167099, −9.14166780283, −8.90953062472, −7.91852369734, −7.39965689908, −6.43061601210, −5.59367240785, −4.36516732197, −3.31536628562, 0,
3.31536628562, 4.36516732197, 5.59367240785, 6.43061601210, 7.39965689908, 7.91852369734, 8.90953062472, 9.14166780283, 9.83202167099, 10.4722640146, 11.0639650707, 11.9501410659, 12.2570319287, 13.0679309715, 13.7169857556, 14.2355808567, 15.2511430739, 15.6372193758, 16.3258496789, 16.8581567364, 17.0312131871, 17.7117129448, 18.2052809452, 18.7836615640, 19.3037507485
