| L(s) = 1 | − 2-s − 1.94·3-s + 4-s + 3.45·5-s + 1.94·6-s − 2.39·7-s − 8-s + 0.794·9-s − 3.45·10-s + 0.977·11-s − 1.94·12-s − 5.92·13-s + 2.39·14-s − 6.73·15-s + 16-s − 3.06·17-s − 0.794·18-s + 5.80·19-s + 3.45·20-s + 4.65·21-s − 0.977·22-s − 1.03·23-s + 1.94·24-s + 6.94·25-s + 5.92·26-s + 4.29·27-s − 2.39·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.12·3-s + 0.5·4-s + 1.54·5-s + 0.795·6-s − 0.903·7-s − 0.353·8-s + 0.264·9-s − 1.09·10-s + 0.294·11-s − 0.562·12-s − 1.64·13-s + 0.639·14-s − 1.73·15-s + 0.250·16-s − 0.743·17-s − 0.187·18-s + 1.33·19-s + 0.772·20-s + 1.01·21-s − 0.208·22-s − 0.216·23-s + 0.397·24-s + 1.38·25-s + 1.16·26-s + 0.826·27-s − 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.94T + 3T^{2} \) |
| 5 | \( 1 - 3.45T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 - 0.977T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 + 1.03T + 23T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 - 0.629T + 37T^{2} \) |
| 41 | \( 1 - 7.85T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 + 7.59T + 53T^{2} \) |
| 59 | \( 1 + 7.49T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 - 2.93T + 67T^{2} \) |
| 71 | \( 1 + 0.869T + 71T^{2} \) |
| 73 | \( 1 + 7.50T + 73T^{2} \) |
| 79 | \( 1 - 9.44T + 79T^{2} \) |
| 83 | \( 1 - 8.38T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.090915282421115052420250397561, −7.02422227212690348750058109553, −6.61117888550986602623918892514, −5.96151198452195992521910441623, −5.34245799691735499126058057940, −4.62021941628679597248735805230, −3.00509096415147494883617681664, −2.37120049747204622310499220763, −1.18577561632782967834307613203, 0,
1.18577561632782967834307613203, 2.37120049747204622310499220763, 3.00509096415147494883617681664, 4.62021941628679597248735805230, 5.34245799691735499126058057940, 5.96151198452195992521910441623, 6.61117888550986602623918892514, 7.02422227212690348750058109553, 8.090915282421115052420250397561
