| L(s) = 1 | + 1.32·2-s + 3-s − 0.244·4-s + 1.32·6-s + 1.72·7-s − 2.97·8-s + 9-s + 3.57·11-s − 0.244·12-s − 3.93·13-s + 2.27·14-s − 3.45·16-s − 6.66·17-s + 1.32·18-s + 4.90·19-s + 1.72·21-s + 4.74·22-s + 3.18·23-s − 2.97·24-s − 5.20·26-s + 27-s − 0.421·28-s + 8.59·29-s + 8.88·31-s + 1.37·32-s + 3.57·33-s − 8.82·34-s + ⋯ |
| L(s) = 1 | + 0.936·2-s + 0.577·3-s − 0.122·4-s + 0.540·6-s + 0.650·7-s − 1.05·8-s + 0.333·9-s + 1.07·11-s − 0.0706·12-s − 1.09·13-s + 0.609·14-s − 0.862·16-s − 1.61·17-s + 0.312·18-s + 1.12·19-s + 0.375·21-s + 1.01·22-s + 0.663·23-s − 0.607·24-s − 1.02·26-s + 0.192·27-s − 0.0795·28-s + 1.59·29-s + 1.59·31-s + 0.243·32-s + 0.623·33-s − 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.608129076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.608129076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 - 3.57T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 + 6.66T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 8.88T + 31T^{2} \) |
| 37 | \( 1 - 6.83T + 37T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 + 4.62T + 43T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 + 1.98T + 61T^{2} \) |
| 67 | \( 1 - 0.398T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 7.76T + 97T^{2} \) |
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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
−8.521074559745573107628165440456, −7.945851643822616352683192211461, −6.83493957601144151304987976664, −6.43088833383543688109963156687, −5.24315394742839543154408522408, −4.56821671782235324445238166832, −4.20278000000514263059543761510, −3.02688779050448447161367116106, −2.41089151413256864669082615307, −0.986776733679991985695914224488,
0.986776733679991985695914224488, 2.41089151413256864669082615307, 3.02688779050448447161367116106, 4.20278000000514263059543761510, 4.56821671782235324445238166832, 5.24315394742839543154408522408, 6.43088833383543688109963156687, 6.83493957601144151304987976664, 7.945851643822616352683192211461, 8.521074559745573107628165440456
