| L(s) = 1 | + 1.98·3-s − 4.20·5-s + 3.15·7-s + 0.929·9-s + 2.48·11-s + 5.88·13-s − 8.34·15-s + 6.04·17-s − 3.86·19-s + 6.25·21-s − 3.63·23-s + 12.7·25-s − 4.10·27-s + 7.96·31-s + 4.92·33-s − 13.2·35-s + 0.128·37-s + 11.6·39-s − 0.728·41-s − 3.11·43-s − 3.91·45-s − 5.13·47-s + 2.95·49-s + 11.9·51-s + 7.59·53-s − 10.4·55-s − 7.65·57-s + ⋯ |
| L(s) = 1 | + 1.14·3-s − 1.88·5-s + 1.19·7-s + 0.309·9-s + 0.749·11-s + 1.63·13-s − 2.15·15-s + 1.46·17-s − 0.885·19-s + 1.36·21-s − 0.758·23-s + 2.54·25-s − 0.789·27-s + 1.43·31-s + 0.857·33-s − 2.24·35-s + 0.0210·37-s + 1.86·39-s − 0.113·41-s − 0.475·43-s − 0.582·45-s − 0.749·47-s + 0.421·49-s + 1.67·51-s + 1.04·53-s − 1.41·55-s − 1.01·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.872929667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.872929667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.98T + 3T^{2} \) |
| 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 - 6.04T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 - 0.128T + 37T^{2} \) |
| 41 | \( 1 + 0.728T + 41T^{2} \) |
| 43 | \( 1 + 3.11T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 7.59T + 53T^{2} \) |
| 59 | \( 1 + 6.07T + 59T^{2} \) |
| 61 | \( 1 - 0.0743T + 61T^{2} \) |
| 67 | \( 1 + 5.89T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 - 9.04T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 2.33T + 83T^{2} \) |
| 89 | \( 1 + 6.63T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.077666220865941465084892445746, −7.73105816770457665576877491003, −6.78216565968788368840835075872, −5.94275564009223646014996343373, −4.81183124074170589277832162123, −4.09599809533728443046971561674, −3.64816792677624409370228124262, −3.04234906874066928950453899877, −1.76700221104589791496095653804, −0.877098902645612183128467691427,
0.877098902645612183128467691427, 1.76700221104589791496095653804, 3.04234906874066928950453899877, 3.64816792677624409370228124262, 4.09599809533728443046971561674, 4.81183124074170589277832162123, 5.94275564009223646014996343373, 6.78216565968788368840835075872, 7.73105816770457665576877491003, 8.077666220865941465084892445746
