| L(s) = 1 | + 1.07·2-s − 3.31·3-s − 0.848·4-s + 2.71·5-s − 3.56·6-s + 4.57·7-s − 3.05·8-s + 8.01·9-s + 2.91·10-s + 0.517·11-s + 2.81·12-s − 13-s + 4.91·14-s − 9.02·15-s − 1.58·16-s − 2.01·17-s + 8.60·18-s + 2.90·19-s − 2.30·20-s − 15.1·21-s + 0.554·22-s − 6.38·23-s + 10.1·24-s + 2.39·25-s − 1.07·26-s − 16.6·27-s − 3.88·28-s + ⋯ |
| L(s) = 1 | + 0.758·2-s − 1.91·3-s − 0.424·4-s + 1.21·5-s − 1.45·6-s + 1.73·7-s − 1.08·8-s + 2.67·9-s + 0.923·10-s + 0.155·11-s + 0.812·12-s − 0.277·13-s + 1.31·14-s − 2.33·15-s − 0.395·16-s − 0.489·17-s + 2.02·18-s + 0.666·19-s − 0.515·20-s − 3.31·21-s + 0.118·22-s − 1.33·23-s + 2.07·24-s + 0.479·25-s − 0.210·26-s − 3.20·27-s − 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.638352856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638352856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.07T + 2T^{2} \) |
| 3 | \( 1 + 3.31T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 - 0.517T + 11T^{2} \) |
| 17 | \( 1 + 2.01T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 + 5.81T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 6.06T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 7.51T + 67T^{2} \) |
| 71 | \( 1 + 1.01T + 71T^{2} \) |
| 73 | \( 1 + 0.972T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 - 0.820T + 83T^{2} \) |
| 89 | \( 1 - 0.671T + 89T^{2} \) |
| 97 | \( 1 - 4.51T + 97T^{2} \) |
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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
−10.01359758815431925196147904547, −8.945288763355319826464680276007, −7.83736227770596408627012112062, −6.70459530004640678471835586842, −5.89233174777319491142383958848, −5.43907331557489391114209045679, −4.73808610406713928823961270793, −4.21213085254028546267581803551, −2.13638625973535484290029270419, −0.982308089758743672658988845884,
0.982308089758743672658988845884, 2.13638625973535484290029270419, 4.21213085254028546267581803551, 4.73808610406713928823961270793, 5.43907331557489391114209045679, 5.89233174777319491142383958848, 6.70459530004640678471835586842, 7.83736227770596408627012112062, 8.945288763355319826464680276007, 10.01359758815431925196147904547
