| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 4·13-s + 16-s − 18-s − 24-s + 25-s − 4·26-s + 27-s − 32-s + 36-s + 4·37-s + 4·39-s + 48-s + 2·49-s − 50-s + 4·52-s − 54-s − 20·61-s + 64-s − 72-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.235·18-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.176·32-s + 1/6·36-s + 0.657·37-s + 0.640·39-s + 0.144·48-s + 2/7·49-s − 0.141·50-s + 0.554·52-s − 0.136·54-s − 2.56·61-s + 1/8·64-s − 0.117·72-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.081141989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.081141989\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.67015888052155601125139797767, −10.39160509435207624542212960871, −9.415401762585890013045544643092, −9.305587122869086271308905422277, −8.713350404322888410914105544855, −7.941231322257043910223245152113, −7.88297907167905508023143292047, −6.96170559755127999142057961261, −6.42217617652666865799421983967, −5.85236328773314391596499497567, −4.97026786036012165020386374880, −4.08471431361169401081275954745, −3.36585804145949552210873743484, −2.50374962358163435732612030839, −1.36243712364701817507861978801,
1.36243712364701817507861978801, 2.50374962358163435732612030839, 3.36585804145949552210873743484, 4.08471431361169401081275954745, 4.97026786036012165020386374880, 5.85236328773314391596499497567, 6.42217617652666865799421983967, 6.96170559755127999142057961261, 7.88297907167905508023143292047, 7.941231322257043910223245152113, 8.713350404322888410914105544855, 9.305587122869086271308905422277, 9.415401762585890013045544643092, 10.39160509435207624542212960871, 10.67015888052155601125139797767
