| L(s) = 1 | + 0.858·2-s − 1.26·4-s − 3.88·7-s − 2.80·8-s + 1.39·11-s − 3.36·13-s − 3.33·14-s + 0.118·16-s + 3.11·17-s − 2.70·19-s + 1.20·22-s − 6.43·23-s − 2.88·26-s + 4.89·28-s − 8.26·29-s − 6.34·31-s + 5.70·32-s + 2.67·34-s + 7.49·37-s − 2.32·38-s − 11.3·41-s + 3.39·43-s − 1.76·44-s − 5.53·46-s + 8.38·47-s + 8.05·49-s + 4.24·52-s + ⋯ |
| L(s) = 1 | + 0.607·2-s − 0.631·4-s − 1.46·7-s − 0.990·8-s + 0.421·11-s − 0.932·13-s − 0.890·14-s + 0.0296·16-s + 0.755·17-s − 0.620·19-s + 0.256·22-s − 1.34·23-s − 0.566·26-s + 0.925·28-s − 1.53·29-s − 1.13·31-s + 1.00·32-s + 0.458·34-s + 1.23·37-s − 0.376·38-s − 1.77·41-s + 0.518·43-s − 0.266·44-s − 0.815·46-s + 1.22·47-s + 1.15·49-s + 0.588·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8632939859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8632939859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.858T + 2T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 + 6.43T + 23T^{2} \) |
| 29 | \( 1 + 8.26T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 - 7.49T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 7.64T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.54T + 67T^{2} \) |
| 71 | \( 1 + 1.18T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 1.40T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 - 1.51T + 97T^{2} \) |
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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
−8.124179235432274982424059459035, −7.30739112859356387197927449649, −6.59190395446623490171129075937, −5.77663329121280869009269653722, −5.44453226078301476663878312332, −4.24018465853235484289750823309, −3.81021937976387087001669197101, −3.07100046114396335347229503427, −2.09402714688877525691158912076, −0.42843405805959211245382015050,
0.42843405805959211245382015050, 2.09402714688877525691158912076, 3.07100046114396335347229503427, 3.81021937976387087001669197101, 4.24018465853235484289750823309, 5.44453226078301476663878312332, 5.77663329121280869009269653722, 6.59190395446623490171129075937, 7.30739112859356387197927449649, 8.124179235432274982424059459035
