| L(s) = 1 | + 2.73·2-s + 5.46·4-s + 0.732·5-s + 3.73·7-s + 9.46·8-s + 2·10-s + 0.267·13-s + 10.1·14-s + 14.9·16-s − 4.19·17-s + 5.19·19-s + 4·20-s − 8·23-s − 4.46·25-s + 0.732·26-s + 20.3·28-s − 1.26·29-s + 0.535·31-s + 21.8·32-s − 11.4·34-s + 2.73·35-s − 6.46·37-s + 14.1·38-s + 6.92·40-s + 1.46·41-s − 3.46·43-s − 21.8·46-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.73·4-s + 0.327·5-s + 1.41·7-s + 3.34·8-s + 0.632·10-s + 0.0743·13-s + 2.72·14-s + 3.73·16-s − 1.01·17-s + 1.19·19-s + 0.894·20-s − 1.66·23-s − 0.892·25-s + 0.143·26-s + 3.85·28-s − 0.235·29-s + 0.0962·31-s + 3.86·32-s − 1.96·34-s + 0.461·35-s − 1.06·37-s + 2.30·38-s + 1.09·40-s + 0.228·41-s − 0.528·43-s − 3.22·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.222478330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.222478330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 13 | \( 1 - 0.267T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 4.26T + 61T^{2} \) |
| 67 | \( 1 - 5.92T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 0.339T + 89T^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−8.200926429652364122672091042867, −7.76962570671786487152308430724, −6.85981541718734233847150399030, −6.13112834876282934301004446392, −5.38271612213209932829614683743, −4.84783780492674273925879374226, −4.11751932242469296594652772007, −3.31658559922733930064782479692, −2.11601101881608347878569705749, −1.67597895253750507910167912655,
1.67597895253750507910167912655, 2.11601101881608347878569705749, 3.31658559922733930064782479692, 4.11751932242469296594652772007, 4.84783780492674273925879374226, 5.38271612213209932829614683743, 6.13112834876282934301004446392, 6.85981541718734233847150399030, 7.76962570671786487152308430724, 8.200926429652364122672091042867
