| L(s) = 1 | + 1.89·2-s − 3.30·3-s + 1.60·4-s + 5-s − 6.26·6-s + 1.63·7-s − 0.744·8-s + 7.89·9-s + 1.89·10-s + 0.589·11-s − 5.30·12-s + 4.93·13-s + 3.10·14-s − 3.30·15-s − 4.63·16-s + 5.34·17-s + 14.9·18-s + 1.04·19-s + 1.60·20-s − 5.38·21-s + 1.12·22-s − 2.14·23-s + 2.45·24-s + 25-s + 9.37·26-s − 16.1·27-s + 2.62·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 1.90·3-s + 0.804·4-s + 0.447·5-s − 2.55·6-s + 0.617·7-s − 0.263·8-s + 2.63·9-s + 0.600·10-s + 0.177·11-s − 1.53·12-s + 1.36·13-s + 0.828·14-s − 0.852·15-s − 1.15·16-s + 1.29·17-s + 3.53·18-s + 0.239·19-s + 0.359·20-s − 1.17·21-s + 0.238·22-s − 0.448·23-s + 0.501·24-s + 0.200·25-s + 1.83·26-s − 3.10·27-s + 0.496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780774856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780774856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 - 0.589T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 - 5.34T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 + 7.99T + 43T^{2} \) |
| 47 | \( 1 + 0.401T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 4.07T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 1.28T + 73T^{2} \) |
| 79 | \( 1 + 2.66T + 79T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−11.65075350053221359994953866754, −10.64049625651573456821241319805, −9.904830164729116558516242115730, −8.310547315054618039705350650310, −6.72991300626215718548959551515, −6.18776914478357664491189116616, −5.32869345268003375147260450367, −4.75849697142234435619822563480, −3.57604152526464664713181607235, −1.33142974151326849745988235147,
1.33142974151326849745988235147, 3.57604152526464664713181607235, 4.75849697142234435619822563480, 5.32869345268003375147260450367, 6.18776914478357664491189116616, 6.72991300626215718548959551515, 8.310547315054618039705350650310, 9.904830164729116558516242115730, 10.64049625651573456821241319805, 11.65075350053221359994953866754
