| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 4·13-s + 16-s − 12·17-s + 2·20-s + 3·25-s + 4·26-s + 12·29-s + 32-s − 12·34-s + 4·37-s + 2·40-s + 12·41-s + 2·49-s + 3·50-s + 4·52-s + 12·53-s + 12·58-s − 20·61-s + 64-s + 8·65-s − 12·68-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 1.10·13-s + 1/4·16-s − 2.91·17-s + 0.447·20-s + 3/5·25-s + 0.784·26-s + 2.22·29-s + 0.176·32-s − 2.05·34-s + 0.657·37-s + 0.316·40-s + 1.87·41-s + 2/7·49-s + 0.424·50-s + 0.554·52-s + 1.64·53-s + 1.57·58-s − 2.56·61-s + 1/8·64-s + 0.992·65-s − 1.45·68-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.588580280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.588580280\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−9.880541566397139490214892173475, −9.415401762585890013045544643092, −8.713350404322888410914105544855, −8.644229019078172602684577487278, −7.88297907167905508023143292047, −6.96170559755127999142057961261, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −5.85236328773314391596499497567, −4.97026786036012165020386374880, −4.32687879913691587957737883725, −4.08471431361169401081275954745, −2.74628747875934588873509462774, −2.50374962358163435732612030839, −1.36243712364701817507861978801,
1.36243712364701817507861978801, 2.50374962358163435732612030839, 2.74628747875934588873509462774, 4.08471431361169401081275954745, 4.32687879913691587957737883725, 4.97026786036012165020386374880, 5.85236328773314391596499497567, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 6.96170559755127999142057961261, 7.88297907167905508023143292047, 8.644229019078172602684577487278, 8.713350404322888410914105544855, 9.415401762585890013045544643092, 9.880541566397139490214892173475
