| L(s) = 1 | − 0.238·3-s − 3.65·5-s − 2.28·7-s − 2.94·9-s − 3.94·11-s − 4.52·13-s + 0.871·15-s + 2·17-s − 2.83·19-s + 0.545·21-s − 5.83·23-s + 8.35·25-s + 1.41·27-s + 0.0496·29-s + 2.92·31-s + 0.939·33-s + 8.36·35-s − 37-s + 1.07·39-s − 10.1·41-s − 3.31·43-s + 10.7·45-s + 13.1·47-s − 1.76·49-s − 0.476·51-s + 0.188·53-s + 14.4·55-s + ⋯ |
| L(s) = 1 | − 0.137·3-s − 1.63·5-s − 0.864·7-s − 0.981·9-s − 1.18·11-s − 1.25·13-s + 0.224·15-s + 0.485·17-s − 0.650·19-s + 0.119·21-s − 1.21·23-s + 1.67·25-s + 0.272·27-s + 0.00922·29-s + 0.524·31-s + 0.163·33-s + 1.41·35-s − 0.164·37-s + 0.172·39-s − 1.58·41-s − 0.504·43-s + 1.60·45-s + 1.91·47-s − 0.252·49-s − 0.0667·51-s + 0.0259·53-s + 1.94·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1447249114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1447249114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 0.238T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 - 0.0496T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 3.31T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 0.188T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 2.38T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 1.62T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.707369945611573809613042531567, −8.172584722528399002503604745382, −7.53094435118923043934063061029, −6.82083268437211152686690968332, −5.79065238039002887148179841499, −4.98789693195949598914403345774, −4.07821126570479654161993598091, −3.18789597874766926951187686693, −2.48719191630239276845013541580, −0.22476075529772346263076210865,
0.22476075529772346263076210865, 2.48719191630239276845013541580, 3.18789597874766926951187686693, 4.07821126570479654161993598091, 4.98789693195949598914403345774, 5.79065238039002887148179841499, 6.82083268437211152686690968332, 7.53094435118923043934063061029, 8.172584722528399002503604745382, 8.707369945611573809613042531567
