| L(s) = 1 | − 2.97·3-s − 3.46·5-s + 0.700·7-s + 5.87·9-s + 11-s + 0.506·13-s + 10.3·15-s + 0.704·17-s + 1.86·19-s − 2.08·21-s + 0.949·23-s + 7.01·25-s − 8.55·27-s + 4.63·29-s + 3.79·31-s − 2.97·33-s − 2.42·35-s − 2.62·37-s − 1.50·39-s + 12.6·41-s − 4.51·43-s − 20.3·45-s + 8.04·47-s − 6.50·49-s − 2.09·51-s − 4.49·53-s − 3.46·55-s + ⋯ |
| L(s) = 1 | − 1.71·3-s − 1.55·5-s + 0.264·7-s + 1.95·9-s + 0.301·11-s + 0.140·13-s + 2.66·15-s + 0.170·17-s + 0.427·19-s − 0.455·21-s + 0.198·23-s + 1.40·25-s − 1.64·27-s + 0.861·29-s + 0.681·31-s − 0.518·33-s − 0.410·35-s − 0.430·37-s − 0.241·39-s + 1.97·41-s − 0.688·43-s − 3.03·45-s + 1.17·47-s − 0.929·49-s − 0.293·51-s − 0.617·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6601631547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6601631547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 0.700T + 7T^{2} \) |
| 13 | \( 1 - 0.506T + 13T^{2} \) |
| 17 | \( 1 - 0.704T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 - 0.949T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 - 3.79T + 31T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 - 8.04T + 47T^{2} \) |
| 53 | \( 1 + 4.49T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 - 6.22T + 83T^{2} \) |
| 89 | \( 1 + 4.96T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.78712019275788977355284908592, −7.39212926959267156314193575728, −6.56175158650251596502713090861, −6.02471456620605784679764910234, −5.08457504877133349738407102046, −4.55340218807420326854556763810, −3.95802905680993562310917113667, −2.97727662541293953593435387423, −1.33624336146166242439109837627, −0.52664042152205109082725652358,
0.52664042152205109082725652358, 1.33624336146166242439109837627, 2.97727662541293953593435387423, 3.95802905680993562310917113667, 4.55340218807420326854556763810, 5.08457504877133349738407102046, 6.02471456620605784679764910234, 6.56175158650251596502713090861, 7.39212926959267156314193575728, 7.78712019275788977355284908592
