| L(s) = 1 | − 3.27·3-s − 3.27·5-s + 1.61·7-s + 7.74·9-s + 1.66·11-s + 13-s + 10.7·15-s − 3.52·17-s + 3.79·19-s − 5.27·21-s + 8.79·23-s + 5.74·25-s − 15.5·27-s + 4.24·29-s − 7.68·31-s − 5.46·33-s − 5.27·35-s − 3.94·37-s − 3.27·39-s − 10.7·41-s − 2.49·43-s − 25.3·45-s + 8.94·47-s − 4.40·49-s + 11.5·51-s + 2.12·53-s − 5.46·55-s + ⋯ |
| L(s) = 1 | − 1.89·3-s − 1.46·5-s + 0.608·7-s + 2.58·9-s + 0.502·11-s + 0.277·13-s + 2.77·15-s − 0.853·17-s + 0.871·19-s − 1.15·21-s + 1.83·23-s + 1.14·25-s − 2.99·27-s + 0.788·29-s − 1.38·31-s − 0.951·33-s − 0.892·35-s − 0.648·37-s − 0.524·39-s − 1.68·41-s − 0.380·43-s − 3.78·45-s + 1.30·47-s − 0.629·49-s + 1.61·51-s + 0.292·53-s − 0.736·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6518204247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6518204247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 3.79T + 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 0.0891T + 67T^{2} \) |
| 71 | \( 1 - 3.05T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 0.292T + 79T^{2} \) |
| 83 | \( 1 - 4.08T + 83T^{2} \) |
| 89 | \( 1 + 1.22T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.576501896157814368217825305473, −7.60217868360547362244817475418, −7.00281142691810937202671180355, −6.54911171765161067202123119661, −5.36340614441759264814095372091, −4.92953021852667789374775455226, −4.19707955859078717954518082039, −3.39350815816958757597952249102, −1.52316606571783540373005831515, −0.57145273129244826273102682874,
0.57145273129244826273102682874, 1.52316606571783540373005831515, 3.39350815816958757597952249102, 4.19707955859078717954518082039, 4.92953021852667789374775455226, 5.36340614441759264814095372091, 6.54911171765161067202123119661, 7.00281142691810937202671180355, 7.60217868360547362244817475418, 8.576501896157814368217825305473
