| L(s) = 1 | + 2-s + 0.803·3-s + 4-s + 2.17·5-s + 0.803·6-s + 3.14·7-s + 8-s − 2.35·9-s + 2.17·10-s + 1.83·11-s + 0.803·12-s − 2.04·13-s + 3.14·14-s + 1.75·15-s + 16-s + 4.31·17-s − 2.35·18-s − 3.93·19-s + 2.17·20-s + 2.53·21-s + 1.83·22-s + 7.44·23-s + 0.803·24-s − 0.249·25-s − 2.04·26-s − 4.30·27-s + 3.14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.464·3-s + 0.5·4-s + 0.974·5-s + 0.328·6-s + 1.19·7-s + 0.353·8-s − 0.784·9-s + 0.689·10-s + 0.554·11-s + 0.232·12-s − 0.567·13-s + 0.841·14-s + 0.452·15-s + 0.250·16-s + 1.04·17-s − 0.554·18-s − 0.902·19-s + 0.487·20-s + 0.552·21-s + 0.391·22-s + 1.55·23-s + 0.164·24-s − 0.0499·25-s − 0.401·26-s − 0.828·27-s + 0.595·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.375982092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.375982092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 0.803T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 + 3.93T + 19T^{2} \) |
| 23 | \( 1 - 7.44T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 37 | \( 1 + 6.86T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 - 4.46T + 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 - 6.75T + 61T^{2} \) |
| 67 | \( 1 + 0.452T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 6.00T + 79T^{2} \) |
| 83 | \( 1 - 3.67T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
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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
−8.044284589395633104974537609713, −7.36143789827294937242683306285, −6.55617767556165884202541215666, −5.71495882012227787352570616323, −5.28661248909355825032791310109, −4.54382573476903158251300160277, −3.62636889701404085393646958986, −2.69263719724445512508812540978, −2.08452200243513372990052168395, −1.16357147965608553772203662663,
1.16357147965608553772203662663, 2.08452200243513372990052168395, 2.69263719724445512508812540978, 3.62636889701404085393646958986, 4.54382573476903158251300160277, 5.28661248909355825032791310109, 5.71495882012227787352570616323, 6.55617767556165884202541215666, 7.36143789827294937242683306285, 8.044284589395633104974537609713
