| L(s) = 1 | − 2·2-s − 4·3-s − 3·5-s + 8·6-s − 6·7-s + 4·8-s + 8·9-s + 6·10-s − 6·11-s − 7·13-s + 12·14-s + 12·15-s − 4·16-s + 17-s − 16·18-s − 11·19-s + 24·21-s + 12·22-s + 3·23-s − 16·24-s + 2·25-s + 14·26-s − 12·27-s − 4·29-s − 24·30-s − 9·31-s + 24·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s − 1.34·5-s + 3.26·6-s − 2.26·7-s + 1.41·8-s + 8/3·9-s + 1.89·10-s − 1.80·11-s − 1.94·13-s + 3.20·14-s + 3.09·15-s − 16-s + 0.242·17-s − 3.77·18-s − 2.52·19-s + 5.23·21-s + 2.55·22-s + 0.625·23-s − 3.26·24-s + 2/5·25-s + 2.74·26-s − 2.30·27-s − 0.742·29-s − 4.38·30-s − 1.61·31-s + 4.17·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| 491 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 22 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 11 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 55 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 92 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T - 25 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 103 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T - 33 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−15.5159037449, −15.0260440158, −14.6956941381, −13.5755929570, −12.9197266895, −12.7846736713, −12.5991133615, −12.1558903899, −11.3888097872, −11.0059139367, −10.5668705787, −10.2430943664, −9.88355414555, −9.24142428674, −9.00343422797, −7.99952201771, −7.58414073538, −7.35158977306, −6.47472381735, −6.22747904714, −5.38676395218, −4.94125922758, −4.35713664629, −3.57803542744, −2.49046230619, 0, 0, 0,
2.49046230619, 3.57803542744, 4.35713664629, 4.94125922758, 5.38676395218, 6.22747904714, 6.47472381735, 7.35158977306, 7.58414073538, 7.99952201771, 9.00343422797, 9.24142428674, 9.88355414555, 10.2430943664, 10.5668705787, 11.0059139367, 11.3888097872, 12.1558903899, 12.5991133615, 12.7846736713, 12.9197266895, 13.5755929570, 14.6956941381, 15.0260440158, 15.5159037449
