| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 2·9-s + 2·14-s + 16-s − 17-s + 2·18-s − 14·23-s − 25-s + 2·28-s + 6·31-s + 32-s − 34-s + 2·36-s − 5·41-s − 14·46-s + 7·47-s + 5·49-s − 50-s + 2·56-s + 6·62-s + 4·63-s + 64-s − 68-s − 26·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 2/3·9-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.471·18-s − 2.91·23-s − 1/5·25-s + 0.377·28-s + 1.07·31-s + 0.176·32-s − 0.171·34-s + 1/3·36-s − 0.780·41-s − 2.06·46-s + 1.02·47-s + 5/7·49-s − 0.141·50-s + 0.267·56-s + 0.762·62-s + 0.503·63-s + 1/8·64-s − 0.121·68-s − 3.08·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962611464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962611464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 191 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.62568836253306071673732054988, −10.17407737261276440003145294717, −9.947135977322009842513373671912, −9.042682854241946217968178648794, −8.425563676112510119340051542023, −7.86765929867489967440726902719, −7.48219687920498682034105288097, −6.72927967112185856729530136877, −6.08297860483442248886341827672, −5.63490496466260502638218294432, −4.72618195601609781932857717580, −4.27399667014652757006015136100, −3.67975111074027991085489251918, −2.48183014565161592351785890256, −1.67329874314093529772851850688,
1.67329874314093529772851850688, 2.48183014565161592351785890256, 3.67975111074027991085489251918, 4.27399667014652757006015136100, 4.72618195601609781932857717580, 5.63490496466260502638218294432, 6.08297860483442248886341827672, 6.72927967112185856729530136877, 7.48219687920498682034105288097, 7.86765929867489967440726902719, 8.425563676112510119340051542023, 9.042682854241946217968178648794, 9.947135977322009842513373671912, 10.17407737261276440003145294717, 10.62568836253306071673732054988
