| L(s) = 1 | + 1.17·3-s + 5-s + 7-s − 1.62·9-s − 0.490·11-s + 13-s + 1.17·15-s − 7.57·17-s + 3.54·19-s + 1.17·21-s − 4.51·23-s + 25-s − 5.42·27-s + 9.36·29-s + 7.29·31-s − 0.575·33-s + 35-s + 11.3·37-s + 1.17·39-s + 11.4·41-s − 6.49·43-s − 1.62·45-s + 6.49·47-s + 49-s − 8.89·51-s + 9.10·53-s − 0.490·55-s + ⋯ |
| L(s) = 1 | + 0.677·3-s + 0.447·5-s + 0.377·7-s − 0.540·9-s − 0.147·11-s + 0.277·13-s + 0.303·15-s − 1.83·17-s + 0.814·19-s + 0.256·21-s − 0.942·23-s + 0.200·25-s − 1.04·27-s + 1.73·29-s + 1.31·31-s − 0.100·33-s + 0.169·35-s + 1.86·37-s + 0.187·39-s + 1.78·41-s − 0.989·43-s − 0.241·45-s + 0.947·47-s + 0.142·49-s − 1.24·51-s + 1.25·53-s − 0.0661·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.583393793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.583393793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 11 | \( 1 + 0.490T + 11T^{2} \) |
| 17 | \( 1 + 7.57T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + 4.51T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 - 9.10T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 - 8.98T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 + 8.28T + 97T^{2} \) |
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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
−8.390096997522863667278313329541, −8.141390428416471476334758970281, −7.06820396774009992809563482099, −6.29597843341692810778441861872, −5.62058023716131726194585272044, −4.61328485668969535754790718918, −3.95491880035417241853818456571, −2.61188113059511580442812722334, −2.40323080787449136596337429545, −0.917435351574040538191319195143,
0.917435351574040538191319195143, 2.40323080787449136596337429545, 2.61188113059511580442812722334, 3.95491880035417241853818456571, 4.61328485668969535754790718918, 5.62058023716131726194585272044, 6.29597843341692810778441861872, 7.06820396774009992809563482099, 8.141390428416471476334758970281, 8.390096997522863667278313329541
