| L(s) = 1 | − 2-s − 4-s + 3·8-s + 9-s − 4·13-s − 16-s + 2·17-s − 18-s + 10·25-s + 4·26-s − 5·32-s − 2·34-s − 36-s + 2·49-s − 10·50-s + 4·52-s + 12·53-s + 7·64-s − 2·68-s + 3·72-s + 81-s + 20·89-s − 2·98-s − 10·100-s + 12·101-s − 12·104-s − 12·106-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1/3·9-s − 1.10·13-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 2·25-s + 0.784·26-s − 0.883·32-s − 0.342·34-s − 1/6·36-s + 2/7·49-s − 1.41·50-s + 0.554·52-s + 1.64·53-s + 7/8·64-s − 0.242·68-s + 0.353·72-s + 1/9·81-s + 2.11·89-s − 0.202·98-s − 100-s + 1.19·101-s − 1.17·104-s − 1.16·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7759443292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7759443292\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−10.20038616398405756967349367575, −9.734494705668299231615576779056, −9.165697416183880804303893132121, −8.808539976636074898519536020356, −8.250678548770432513372751539390, −7.64269603874240746157312204999, −7.19530210990747938268153493228, −6.76504709735163753052933191320, −5.86397751404295402847855963607, −5.08231243962829380697245103476, −4.78291136122247326786141787105, −4.02188122031326001640845020005, −3.15907665366773433027442605075, −2.18518691870592382653137399729, −0.927303156203294391380959526336,
0.927303156203294391380959526336, 2.18518691870592382653137399729, 3.15907665366773433027442605075, 4.02188122031326001640845020005, 4.78291136122247326786141787105, 5.08231243962829380697245103476, 5.86397751404295402847855963607, 6.76504709735163753052933191320, 7.19530210990747938268153493228, 7.64269603874240746157312204999, 8.250678548770432513372751539390, 8.808539976636074898519536020356, 9.165697416183880804303893132121, 9.734494705668299231615576779056, 10.20038616398405756967349367575
