L(s) = 1 | + 2-s + 4-s + 8-s + 16-s − 6·19-s − 4·25-s + 8·29-s + 32-s − 6·38-s − 16·41-s + 12·43-s + 2·49-s − 4·50-s + 8·53-s + 8·58-s + 8·59-s + 12·61-s + 64-s + 16·71-s + 16·73-s − 6·76-s − 16·82-s + 12·86-s + 2·98-s − 4·100-s + 8·106-s − 8·107-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s − 1.37·19-s − 4/5·25-s + 1.48·29-s + 0.176·32-s − 0.973·38-s − 2.49·41-s + 1.82·43-s + 2/7·49-s − 0.565·50-s + 1.09·53-s + 1.05·58-s + 1.04·59-s + 1.53·61-s + 1/8·64-s + 1.89·71-s + 1.87·73-s − 0.688·76-s − 1.76·82-s + 1.29·86-s + 0.202·98-s − 2/5·100-s + 0.777·106-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.814410379\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.814410379\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
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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
−8.449214706675140374577137305507, −8.163799071418858750785983823329, −7.66178955572243648409098806269, −6.93273577938547297289305636557, −6.67161708699562132524363177791, −6.37452483206806356490987912829, −5.59453942721608634726857267402, −5.32840831471856328566334839582, −4.75211834002268501330498127169, −4.07007049787849789478087703396, −3.87857022401287507201063255664, −3.11735200948967478070120539529, −2.35304085693774311916549110742, −2.00245026895425214068587158329, −0.800785425372416660468654362251,
0.800785425372416660468654362251, 2.00245026895425214068587158329, 2.35304085693774311916549110742, 3.11735200948967478070120539529, 3.87857022401287507201063255664, 4.07007049787849789478087703396, 4.75211834002268501330498127169, 5.32840831471856328566334839582, 5.59453942721608634726857267402, 6.37452483206806356490987912829, 6.67161708699562132524363177791, 6.93273577938547297289305636557, 7.66178955572243648409098806269, 8.163799071418858750785983823329, 8.449214706675140374577137305507