| L(s) = 1 | + 5-s − 0.761·7-s + 2.86·11-s + 0.864·13-s − 0.761·17-s + 6.38·19-s − 23-s + 25-s + 1.62·29-s − 3.62·31-s − 0.761·35-s + 2.76·37-s + 7.62·41-s + 1.72·43-s + 10.3·47-s − 6.42·49-s − 0.761·53-s + 2.86·55-s − 3.35·59-s − 2.11·61-s + 0.864·65-s − 7.74·67-s − 13.9·71-s + 4.92·73-s − 2.18·77-s + 10.5·79-s − 0.490·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.287·7-s + 0.863·11-s + 0.239·13-s − 0.184·17-s + 1.46·19-s − 0.208·23-s + 0.200·25-s + 0.301·29-s − 0.651·31-s − 0.128·35-s + 0.453·37-s + 1.19·41-s + 0.263·43-s + 1.51·47-s − 0.917·49-s − 0.104·53-s + 0.386·55-s − 0.436·59-s − 0.271·61-s + 0.107·65-s − 0.945·67-s − 1.65·71-s + 0.576·73-s − 0.248·77-s + 1.18·79-s − 0.0538·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488632529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488632529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 0.761T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 - 0.864T + 13T^{2} \) |
| 17 | \( 1 + 0.761T + 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 3.62T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 - 7.62T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 0.761T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 0.490T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 6.77T + 97T^{2} \) |
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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
−7.57678701057787920426637640174, −7.25743275812406750513092805871, −6.21551943475401613350253780204, −5.95363595535186813464035634779, −5.03795435440640961664513496202, −4.24875028426973949862113138864, −3.46596337953633871809561429491, −2.71559181380713158830666389591, −1.68497910534793668716666021733, −0.814168308054472392000466319306,
0.814168308054472392000466319306, 1.68497910534793668716666021733, 2.71559181380713158830666389591, 3.46596337953633871809561429491, 4.24875028426973949862113138864, 5.03795435440640961664513496202, 5.95363595535186813464035634779, 6.21551943475401613350253780204, 7.25743275812406750513092805871, 7.57678701057787920426637640174
