| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 2·9-s − 5·11-s + 2·14-s + 16-s − 2·18-s + 5·22-s + 14·23-s − 6·25-s − 2·28-s − 9·29-s − 32-s + 2·36-s − 3·37-s + 9·43-s − 5·44-s − 14·46-s − 3·49-s + 6·50-s + 9·53-s + 2·56-s + 9·58-s − 4·63-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 2/3·9-s − 1.50·11-s + 0.534·14-s + 1/4·16-s − 0.471·18-s + 1.06·22-s + 2.91·23-s − 6/5·25-s − 0.377·28-s − 1.67·29-s − 0.176·32-s + 1/3·36-s − 0.493·37-s + 1.37·43-s − 0.753·44-s − 2.06·46-s − 3/7·49-s + 0.848·50-s + 1.23·53-s + 0.267·56-s + 1.18·58-s − 0.503·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8920653554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8920653554\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 127 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
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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
−9.090733850380420084951653522737, −8.872767875370114843830577502336, −8.092918001546453973956682017131, −7.65587749866278308407010116721, −7.24486035699223528766675329636, −6.96353645848359802219899303546, −6.30170507405223240178984721566, −5.61042384745585683353798748324, −5.30267265140376940406222916927, −4.65175192499576485855840994116, −3.80922609847443711098275479928, −3.23546949660176218352964183948, −2.60623042047293500918612996904, −1.87727852621484565949770037151, −0.66958571965107075055353006393,
0.66958571965107075055353006393, 1.87727852621484565949770037151, 2.60623042047293500918612996904, 3.23546949660176218352964183948, 3.80922609847443711098275479928, 4.65175192499576485855840994116, 5.30267265140376940406222916927, 5.61042384745585683353798748324, 6.30170507405223240178984721566, 6.96353645848359802219899303546, 7.24486035699223528766675329636, 7.65587749866278308407010116721, 8.092918001546453973956682017131, 8.872767875370114843830577502336, 9.090733850380420084951653522737
