| L(s) = 1 | + 2.79·3-s − 2·5-s + 3.79·7-s + 4.79·9-s + 11-s + 13-s − 5.58·15-s − 8.37·19-s + 10.5·21-s + 0.208·23-s − 25-s + 4.99·27-s + 7.58·29-s + 4·31-s + 2.79·33-s − 7.58·35-s + 8·37-s + 2.79·39-s − 5.79·41-s + 8·43-s − 9.58·45-s − 11.5·47-s + 7.37·49-s − 9.95·53-s − 2·55-s − 23.3·57-s − 9.58·59-s + ⋯ |
| L(s) = 1 | + 1.61·3-s − 0.894·5-s + 1.43·7-s + 1.59·9-s + 0.301·11-s + 0.277·13-s − 1.44·15-s − 1.92·19-s + 2.30·21-s + 0.0435·23-s − 0.200·25-s + 0.962·27-s + 1.40·29-s + 0.718·31-s + 0.485·33-s − 1.28·35-s + 1.31·37-s + 0.446·39-s − 0.904·41-s + 1.21·43-s − 1.42·45-s − 1.68·47-s + 1.05·49-s − 1.36·53-s − 0.269·55-s − 3.09·57-s − 1.24·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.440584819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.440584819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 8.37T + 19T^{2} \) |
| 23 | \( 1 - 0.208T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 9.95T + 53T^{2} \) |
| 59 | \( 1 + 9.58T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 - 3.37T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−10.77732603378243054948540143618, −9.674288166903585298252103594589, −8.457511725447752653649768741344, −8.353990272065340725122609113527, −7.60870808654391371596862613885, −6.42619738798258669011493112896, −4.59695211239201614821122891649, −4.10863107379942512053190516488, −2.83239478512247258608839892656, −1.65634084547744042653169411871,
1.65634084547744042653169411871, 2.83239478512247258608839892656, 4.10863107379942512053190516488, 4.59695211239201614821122891649, 6.42619738798258669011493112896, 7.60870808654391371596862613885, 8.353990272065340725122609113527, 8.457511725447752653649768741344, 9.674288166903585298252103594589, 10.77732603378243054948540143618
