| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 6·7-s − 8-s − 2·9-s − 12-s − 6·14-s + 16-s + 2·18-s − 6·21-s + 24-s − 2·25-s + 5·27-s + 6·28-s + 10·29-s − 32-s − 2·36-s + 8·41-s + 6·42-s − 12·43-s − 48-s + 17·49-s + 2·50-s − 2·53-s − 5·54-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 2.26·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s − 1.60·14-s + 1/4·16-s + 0.471·18-s − 1.30·21-s + 0.204·24-s − 2/5·25-s + 0.962·27-s + 1.13·28-s + 1.85·29-s − 0.176·32-s − 1/3·36-s + 1.24·41-s + 0.925·42-s − 1.82·43-s − 0.144·48-s + 17/7·49-s + 0.282·50-s − 0.274·53-s − 0.680·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.002343927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.002343927\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 75 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.07113220366631324213557733826, −9.653830813477996218717673657091, −8.873193591013862940682562735246, −8.379818514815940464452750097868, −8.084215421719444367670114548405, −7.81913959227647380831995290089, −6.82267782243031164846670210174, −6.56087406329620377147621070166, −5.65022704699238663509395586647, −5.19815433970855149034522267235, −4.75813358390011036794135778335, −3.99371376987675738529571241648, −2.84360916951692733726955331621, −2.02605892764822334144730536655, −1.06391675847599943818182876252,
1.06391675847599943818182876252, 2.02605892764822334144730536655, 2.84360916951692733726955331621, 3.99371376987675738529571241648, 4.75813358390011036794135778335, 5.19815433970855149034522267235, 5.65022704699238663509395586647, 6.56087406329620377147621070166, 6.82267782243031164846670210174, 7.81913959227647380831995290089, 8.084215421719444367670114548405, 8.379818514815940464452750097868, 8.873193591013862940682562735246, 9.653830813477996218717673657091, 10.07113220366631324213557733826
