| L(s) = 1 | + 3-s − 6·11-s + 4·13-s − 4·19-s − 6·23-s + 5·25-s − 27-s + 12·29-s + 8·31-s − 6·33-s − 2·37-s + 4·39-s + 24·41-s + 8·43-s + 12·47-s + 6·53-s − 4·57-s + 10·61-s + 8·67-s − 6·69-s − 12·71-s + 10·73-s + 5·75-s − 4·79-s − 81-s + 24·83-s + 12·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.80·11-s + 1.10·13-s − 0.917·19-s − 1.25·23-s + 25-s − 0.192·27-s + 2.22·29-s + 1.43·31-s − 1.04·33-s − 0.328·37-s + 0.640·39-s + 3.74·41-s + 1.21·43-s + 1.75·47-s + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.977·67-s − 0.722·69-s − 1.42·71-s + 1.17·73-s + 0.577·75-s − 0.450·79-s − 1/9·81-s + 2.63·83-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.263724401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.263724401\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.948570264024871620121846425338, −8.778935401036831239947267251023, −8.373679238947412289545268438251, −8.017069253397462394286008874360, −7.83923337769409244194363015835, −7.43304609625792039376836339982, −6.73037548610876755722401835656, −6.58070181055153680044381197078, −5.93615149394874140145709010498, −5.75848343351756164234287376954, −5.32447292644377262542449593788, −4.65574677400007537791932302968, −4.24858645066785721866275906937, −4.10081317745311599115166427115, −3.37804046707988641401349354688, −2.64498024173241659152332272215, −2.55499019634426510586029306901, −2.27278853436506697571150202001, −1.04903925782933709581693766877, −0.71040944456387151011724551109,
0.71040944456387151011724551109, 1.04903925782933709581693766877, 2.27278853436506697571150202001, 2.55499019634426510586029306901, 2.64498024173241659152332272215, 3.37804046707988641401349354688, 4.10081317745311599115166427115, 4.24858645066785721866275906937, 4.65574677400007537791932302968, 5.32447292644377262542449593788, 5.75848343351756164234287376954, 5.93615149394874140145709010498, 6.58070181055153680044381197078, 6.73037548610876755722401835656, 7.43304609625792039376836339982, 7.83923337769409244194363015835, 8.017069253397462394286008874360, 8.373679238947412289545268438251, 8.778935401036831239947267251023, 8.948570264024871620121846425338
