| L(s) = 1 | + 1.32·2-s + 1.56·3-s − 0.231·4-s + 5-s + 2.07·6-s − 3.50·7-s − 2.96·8-s − 0.561·9-s + 1.32·10-s − 0.659·11-s − 0.362·12-s + 5.91·13-s − 4.65·14-s + 1.56·15-s − 3.48·16-s + 0.844·17-s − 0.746·18-s + 0.659·19-s − 0.231·20-s − 5.47·21-s − 0.876·22-s − 23-s − 4.63·24-s + 25-s + 7.85·26-s − 5.56·27-s + 0.812·28-s + ⋯ |
| L(s) = 1 | + 0.940·2-s + 0.901·3-s − 0.115·4-s + 0.447·5-s + 0.847·6-s − 1.32·7-s − 1.04·8-s − 0.187·9-s + 0.420·10-s − 0.198·11-s − 0.104·12-s + 1.63·13-s − 1.24·14-s + 0.403·15-s − 0.870·16-s + 0.204·17-s − 0.176·18-s + 0.151·19-s − 0.0518·20-s − 1.19·21-s − 0.186·22-s − 0.208·23-s − 0.945·24-s + 0.200·25-s + 1.54·26-s − 1.07·27-s + 0.153·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.712218692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.712218692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 + 0.659T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 - 0.844T + 17T^{2} \) |
| 19 | \( 1 - 0.659T + 19T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 + 9.47T + 43T^{2} \) |
| 47 | \( 1 - 6.88T + 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 - 5.78T + 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 + 3.63T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 + 8.27T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 8.34T + 97T^{2} \) |
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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
−13.57963036176141021296510619764, −13.04634885074001251790285594339, −11.88144176656244784495546319528, −10.27472711198260801204894879963, −9.175151842257403588921190550651, −8.437168023621527615153667015029, −6.53762544700321598932036337105, −5.61870438256016155150094705080, −3.79699061686296646681914444342, −2.93276387909680137409825122262,
2.93276387909680137409825122262, 3.79699061686296646681914444342, 5.61870438256016155150094705080, 6.53762544700321598932036337105, 8.437168023621527615153667015029, 9.175151842257403588921190550651, 10.27472711198260801204894879963, 11.88144176656244784495546319528, 13.04634885074001251790285594339, 13.57963036176141021296510619764
