| L(s) = 1 | − 2·7-s − 4·11-s − 5·13-s − 4·17-s + 8·19-s + 4·23-s + 5·25-s − 8·29-s + 6·31-s + 6·37-s − 12·41-s − 43-s + 6·47-s − 11·49-s + 4·53-s − 10·59-s + 13·61-s + 11·67-s − 6·71-s + 11·73-s + 8·77-s + 79-s − 6·89-s + 10·91-s − 2·97-s + 14·101-s − 26·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s − 1.38·13-s − 0.970·17-s + 1.83·19-s + 0.834·23-s + 25-s − 1.48·29-s + 1.07·31-s + 0.986·37-s − 1.87·41-s − 0.152·43-s + 0.875·47-s − 1.57·49-s + 0.549·53-s − 1.30·59-s + 1.66·61-s + 1.34·67-s − 0.712·71-s + 1.28·73-s + 0.911·77-s + 0.112·79-s − 0.635·89-s + 1.04·91-s − 0.203·97-s + 1.39·101-s − 2.56·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9036267692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9036267692\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.124362721225561214403692272046, −8.702626834706151081135808636067, −8.082392866029877861500992519325, −7.84916955288911936803394069730, −7.64148185930982868331252728535, −6.87972689111588910568510461605, −6.80599851937037139546779841251, −6.66949586248055424210420187394, −5.82593216827826283901988619819, −5.29652923397125957659093818458, −5.26194273546154851020334709692, −4.87183394052702909708307565178, −4.32737447084726215117584665988, −3.78632877542936946812539949477, −3.19947756374256917372779807852, −2.81957979243773189861096770790, −2.60558213526454230453056630330, −1.93055082428825596053398305041, −1.14091470839979226297207057656, −0.32873072987199713083536558795,
0.32873072987199713083536558795, 1.14091470839979226297207057656, 1.93055082428825596053398305041, 2.60558213526454230453056630330, 2.81957979243773189861096770790, 3.19947756374256917372779807852, 3.78632877542936946812539949477, 4.32737447084726215117584665988, 4.87183394052702909708307565178, 5.26194273546154851020334709692, 5.29652923397125957659093818458, 5.82593216827826283901988619819, 6.66949586248055424210420187394, 6.80599851937037139546779841251, 6.87972689111588910568510461605, 7.64148185930982868331252728535, 7.84916955288911936803394069730, 8.082392866029877861500992519325, 8.702626834706151081135808636067, 9.124362721225561214403692272046
