| L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·15-s − 25-s + 27-s + 2·45-s + 10·49-s + 20·53-s − 75-s + 16·79-s + 81-s − 8·83-s − 8·107-s + 18·121-s − 12·125-s + 127-s + 131-s + 2·135-s + 137-s + 139-s + 10·147-s + 149-s + 151-s + 157-s + 20·159-s + 163-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.516·15-s − 1/5·25-s + 0.192·27-s + 0.298·45-s + 10/7·49-s + 2.74·53-s − 0.115·75-s + 1.80·79-s + 1/9·81-s − 0.878·83-s − 0.773·107-s + 1.63·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.172·135-s + 0.0854·137-s + 0.0848·139-s + 0.824·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.58·159-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.752819254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.752819254\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.804232518711272914975833640674, −8.339729437170921920814564547387, −7.88588785443659664968877050779, −7.29400030072680013487098644789, −6.95462669840527913917302121355, −6.43993342010759876282540261240, −5.72009755866980912480255592736, −5.60378398018998027832781220942, −4.89860843972320382069008847636, −4.21523796332319060680877486200, −3.79343751691397527212267079179, −3.06378196949189061119331802386, −2.37025221506316933204103603555, −1.96518584136898307589002473995, −0.950269898347283123135741361773,
0.950269898347283123135741361773, 1.96518584136898307589002473995, 2.37025221506316933204103603555, 3.06378196949189061119331802386, 3.79343751691397527212267079179, 4.21523796332319060680877486200, 4.89860843972320382069008847636, 5.60378398018998027832781220942, 5.72009755866980912480255592736, 6.43993342010759876282540261240, 6.95462669840527913917302121355, 7.29400030072680013487098644789, 7.88588785443659664968877050779, 8.339729437170921920814564547387, 8.804232518711272914975833640674
