| L(s) = 1 | + 2.22·3-s − 4.18·5-s − 0.570·7-s + 1.95·9-s − 0.473·11-s − 0.269·13-s − 9.31·15-s − 2.85·17-s − 1.04·19-s − 1.26·21-s − 8.61·23-s + 12.4·25-s − 2.32·27-s − 0.0972·29-s − 4.42·31-s − 1.05·33-s + 2.38·35-s + 4.22·37-s − 0.600·39-s − 9.67·41-s − 0.946·43-s − 8.18·45-s − 8.25·47-s − 6.67·49-s − 6.36·51-s + 1.08·53-s + 1.97·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 1.87·5-s − 0.215·7-s + 0.652·9-s − 0.142·11-s − 0.0748·13-s − 2.40·15-s − 0.693·17-s − 0.239·19-s − 0.277·21-s − 1.79·23-s + 2.49·25-s − 0.447·27-s − 0.0180·29-s − 0.793·31-s − 0.183·33-s + 0.403·35-s + 0.694·37-s − 0.0962·39-s − 1.51·41-s − 0.144·43-s − 1.21·45-s − 1.20·47-s − 0.953·49-s − 0.891·51-s + 0.149·53-s + 0.266·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 7 | \( 1 + 0.570T + 7T^{2} \) |
| 11 | \( 1 + 0.473T + 11T^{2} \) |
| 13 | \( 1 + 0.269T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 23 | \( 1 + 8.61T + 23T^{2} \) |
| 29 | \( 1 + 0.0972T + 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 + 0.946T + 43T^{2} \) |
| 47 | \( 1 + 8.25T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 - 6.35T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−9.380504436116473854535857372817, −8.462389405923698256162313115799, −8.091461724269971427524106378034, −7.40108037337879531411539373688, −6.44402599967673618578179383398, −4.84722791856217090025254412781, −3.86057666596427874811204502779, −3.39416850551708371573828865018, −2.16977680580925116488168180512, 0,
2.16977680580925116488168180512, 3.39416850551708371573828865018, 3.86057666596427874811204502779, 4.84722791856217090025254412781, 6.44402599967673618578179383398, 7.40108037337879531411539373688, 8.091461724269971427524106378034, 8.462389405923698256162313115799, 9.380504436116473854535857372817
