| L(s) = 1 | − 2·2-s + 4-s − 5-s − 4·9-s + 2·10-s + 11-s + 3·13-s + 16-s + 8·18-s + 2·19-s − 20-s − 2·22-s + 23-s + 5·25-s − 6·26-s − 4·29-s − 6·31-s + 2·32-s − 4·36-s − 10·37-s − 4·38-s − 3·41-s + 8·43-s + 44-s + 4·45-s − 2·46-s − 6·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.447·5-s − 4/3·9-s + 0.632·10-s + 0.301·11-s + 0.832·13-s + 1/4·16-s + 1.88·18-s + 0.458·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 25-s − 1.17·26-s − 0.742·29-s − 1.07·31-s + 0.353·32-s − 2/3·36-s − 1.64·37-s − 0.648·38-s − 0.468·41-s + 1.21·43-s + 0.150·44-s + 0.596·45-s − 0.294·46-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1899303686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1899303686\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 5 T - 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 50 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−19.7584408161, −19.0417248100, −18.5531796814, −18.1468454152, −17.3389199371, −17.0838002888, −16.3856158183, −15.7051202259, −14.9330210292, −14.2544394695, −13.6313060333, −12.6120433993, −11.8661381446, −11.0966840919, −10.6271890885, −9.43105940781, −8.96729298284, −8.42134292864, −7.61748176151, −6.43565236949, −5.29476270386, −3.45243698121,
3.45243698121, 5.29476270386, 6.43565236949, 7.61748176151, 8.42134292864, 8.96729298284, 9.43105940781, 10.6271890885, 11.0966840919, 11.8661381446, 12.6120433993, 13.6313060333, 14.2544394695, 14.9330210292, 15.7051202259, 16.3856158183, 17.0838002888, 17.3389199371, 18.1468454152, 18.5531796814, 19.0417248100, 19.7584408161
