| L(s) = 1 | + 3-s − 3.16·5-s − 0.230·7-s + 9-s + 0.534·11-s + 2.40·13-s − 3.16·15-s + 3.83·17-s − 7.74·19-s − 0.230·21-s + 5.14·23-s + 5.03·25-s + 27-s + 0.602·29-s − 4.43·31-s + 0.534·33-s + 0.731·35-s + 8.52·37-s + 2.40·39-s + 1.40·41-s − 3.66·43-s − 3.16·45-s + 12.4·47-s − 6.94·49-s + 3.83·51-s + 4.85·53-s − 1.69·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.41·5-s − 0.0872·7-s + 0.333·9-s + 0.161·11-s + 0.666·13-s − 0.818·15-s + 0.930·17-s − 1.77·19-s − 0.0503·21-s + 1.07·23-s + 1.00·25-s + 0.192·27-s + 0.111·29-s − 0.797·31-s + 0.0931·33-s + 0.123·35-s + 1.40·37-s + 0.385·39-s + 0.219·41-s − 0.559·43-s − 0.472·45-s + 1.82·47-s − 0.992·49-s + 0.537·51-s + 0.667·53-s − 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630606241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630606241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 + 0.230T + 7T^{2} \) |
| 11 | \( 1 - 0.534T + 11T^{2} \) |
| 13 | \( 1 - 2.40T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 + 7.74T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 - 0.602T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 - 8.52T + 37T^{2} \) |
| 41 | \( 1 - 1.40T + 41T^{2} \) |
| 43 | \( 1 + 3.66T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 4.85T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 + 1.49T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 4.16T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 - 9.40T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.892496936456321811591115439007, −8.383361856065789673523179011937, −7.71296436814910536956418745344, −6.99664514808303411914595954469, −6.11838157831433437381599428906, −4.91104854293450298583373022071, −3.96098715643138789774465634809, −3.53631496106654444544715941335, −2.36453132292340419943923435627, −0.836351500873162869364529352077,
0.836351500873162869364529352077, 2.36453132292340419943923435627, 3.53631496106654444544715941335, 3.96098715643138789774465634809, 4.91104854293450298583373022071, 6.11838157831433437381599428906, 6.99664514808303411914595954469, 7.71296436814910536956418745344, 8.383361856065789673523179011937, 8.892496936456321811591115439007
