| L(s) = 1 | − 2-s + 2.35·3-s + 4-s + 2.64·5-s − 2.35·6-s + 2.15·7-s − 8-s + 2.54·9-s − 2.64·10-s + 4.62·11-s + 2.35·12-s − 5.72·13-s − 2.15·14-s + 6.23·15-s + 16-s + 3.70·17-s − 2.54·18-s − 3.06·19-s + 2.64·20-s + 5.07·21-s − 4.62·22-s + 5.68·23-s − 2.35·24-s + 2.01·25-s + 5.72·26-s − 1.08·27-s + 2.15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s + 1.18·5-s − 0.960·6-s + 0.815·7-s − 0.353·8-s + 0.847·9-s − 0.837·10-s + 1.39·11-s + 0.679·12-s − 1.58·13-s − 0.576·14-s + 1.61·15-s + 0.250·16-s + 0.898·17-s − 0.598·18-s − 0.702·19-s + 0.592·20-s + 1.10·21-s − 0.985·22-s + 1.18·23-s − 0.480·24-s + 0.403·25-s + 1.12·26-s − 0.207·27-s + 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.360875637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.360875637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 0.173T + 31T^{2} \) |
| 37 | \( 1 - 6.11T + 37T^{2} \) |
| 41 | \( 1 + 0.562T + 41T^{2} \) |
| 43 | \( 1 + 13.0T + 43T^{2} \) |
| 47 | \( 1 + 0.0635T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 7.29T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 1.81T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 + 7.44T + 97T^{2} \) |
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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
−8.437117831576289590731458125080, −8.019420185586638519181774687501, −7.07407883951479956613012753892, −6.56617522545857530709721168056, −5.46541001811551607222579194407, −4.68411034324271028609454236752, −3.56463537882062532391485422113, −2.61164343655545801975900354690, −2.00619948422279289255404064536, −1.21187290607500102131659849641,
1.21187290607500102131659849641, 2.00619948422279289255404064536, 2.61164343655545801975900354690, 3.56463537882062532391485422113, 4.68411034324271028609454236752, 5.46541001811551607222579194407, 6.56617522545857530709721168056, 7.07407883951479956613012753892, 8.019420185586638519181774687501, 8.437117831576289590731458125080
