L(s) = 1 | + 3·3-s − 2·4-s + 4·7-s + 6·9-s − 6·12-s + 5·13-s + 6·19-s + 12·21-s − 10·25-s + 9·27-s − 8·28-s − 12·36-s − 10·37-s + 15·39-s − 13·43-s + 9·49-s − 10·52-s + 18·57-s − 15·61-s + 24·63-s + 8·64-s − 11·67-s − 30·75-s − 12·76-s − 26·79-s + 9·81-s − 24·84-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 4-s + 1.51·7-s + 2·9-s − 1.73·12-s + 1.38·13-s + 1.37·19-s + 2.61·21-s − 2·25-s + 1.73·27-s − 1.51·28-s − 2·36-s − 1.64·37-s + 2.40·39-s − 1.98·43-s + 9/7·49-s − 1.38·52-s + 2.38·57-s − 1.92·61-s + 3.02·63-s + 64-s − 1.34·67-s − 3.46·75-s − 1.37·76-s − 2.92·79-s + 81-s − 2.61·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.606837235\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.606837235\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) |
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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
−12.01775081598151894906884131328, −11.64649395896150823072218025337, −11.37501183017857877846891693132, −10.48381003860056307863465221018, −10.16641845627850680700476578894, −9.628603214583999606269110785913, −8.936973863531753034304449028783, −8.925950451305429702959682283376, −8.253558633798219654478234338012, −8.082943810695686673275596558862, −7.45513720265543218143103192689, −7.08623592823649647770118944379, −6.01882596893972613950782762000, −5.48773395311062042178303329598, −4.61041336590011580804505857709, −4.44984241155225366877211282197, −3.42907042790280078111511858902, −3.37554555061242399061086779178, −1.94807609762442698936696248302, −1.51455789003945511277452601152,
1.51455789003945511277452601152, 1.94807609762442698936696248302, 3.37554555061242399061086779178, 3.42907042790280078111511858902, 4.44984241155225366877211282197, 4.61041336590011580804505857709, 5.48773395311062042178303329598, 6.01882596893972613950782762000, 7.08623592823649647770118944379, 7.45513720265543218143103192689, 8.082943810695686673275596558862, 8.253558633798219654478234338012, 8.925950451305429702959682283376, 8.936973863531753034304449028783, 9.628603214583999606269110785913, 10.16641845627850680700476578894, 10.48381003860056307863465221018, 11.37501183017857877846891693132, 11.64649395896150823072218025337, 12.01775081598151894906884131328