| L(s) = 1 | − 2-s + 2.84·3-s + 4-s + 3.13·5-s − 2.84·6-s + 2.34·7-s − 8-s + 5.07·9-s − 3.13·10-s + 3.86·11-s + 2.84·12-s + 1.29·13-s − 2.34·14-s + 8.91·15-s + 16-s + 0.527·17-s − 5.07·18-s − 7.25·19-s + 3.13·20-s + 6.67·21-s − 3.86·22-s − 8.15·23-s − 2.84·24-s + 4.83·25-s − 1.29·26-s + 5.91·27-s + 2.34·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.64·3-s + 0.5·4-s + 1.40·5-s − 1.16·6-s + 0.887·7-s − 0.353·8-s + 1.69·9-s − 0.991·10-s + 1.16·11-s + 0.820·12-s + 0.359·13-s − 0.627·14-s + 2.30·15-s + 0.250·16-s + 0.127·17-s − 1.19·18-s − 1.66·19-s + 0.701·20-s + 1.45·21-s − 0.823·22-s − 1.70·23-s − 0.580·24-s + 0.966·25-s − 0.254·26-s + 1.13·27-s + 0.443·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.940037127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.940037127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 - 3.13T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 - 0.527T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 + 8.15T + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 - 6.03T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 1.54T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 - 0.488T + 79T^{2} \) |
| 83 | \( 1 + 1.15T + 83T^{2} \) |
| 89 | \( 1 - 0.589T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.753836473231651116318696558520, −7.80026366961344764839200033313, −7.39845284713537115599620396816, −6.14554931223266778855553275485, −5.90260564884272462973807559930, −4.30917580960372531880335626837, −3.82641333268919596235701145990, −2.42183831795319659576552166845, −2.05413838037199507151512723272, −1.35836734056887818775743114609,
1.35836734056887818775743114609, 2.05413838037199507151512723272, 2.42183831795319659576552166845, 3.82641333268919596235701145990, 4.30917580960372531880335626837, 5.90260564884272462973807559930, 6.14554931223266778855553275485, 7.39845284713537115599620396816, 7.80026366961344764839200033313, 8.753836473231651116318696558520
