| L(s) = 1 | + 2.67·2-s − 2.97·3-s + 5.16·4-s + 5-s − 7.96·6-s + 4.32·7-s + 8.46·8-s + 5.86·9-s + 2.67·10-s + 3.10·11-s − 15.3·12-s − 4.66·13-s + 11.5·14-s − 2.97·15-s + 12.3·16-s + 2.73·17-s + 15.7·18-s + 3.15·19-s + 5.16·20-s − 12.8·21-s + 8.30·22-s − 6.86·23-s − 25.2·24-s + 25-s − 12.4·26-s − 8.54·27-s + 22.3·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 1.71·3-s + 2.58·4-s + 0.447·5-s − 3.25·6-s + 1.63·7-s + 2.99·8-s + 1.95·9-s + 0.846·10-s + 0.935·11-s − 4.43·12-s − 1.29·13-s + 3.09·14-s − 0.768·15-s + 3.08·16-s + 0.662·17-s + 3.70·18-s + 0.724·19-s + 1.15·20-s − 2.81·21-s + 1.76·22-s − 1.43·23-s − 5.14·24-s + 0.200·25-s − 2.45·26-s − 1.64·27-s + 4.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.603165885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.603165885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 + 2.97T + 3T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 + 7.71T + 29T^{2} \) |
| 31 | \( 1 + 9.06T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 2.37T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 - 0.290T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 0.481T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 97 | \( 1 + 1.72T + 97T^{2} \) |
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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
−9.520489481500963167553112831610, −7.66523349965098119975395461269, −7.33078417007987865061146672293, −6.32636697898091148536738662752, −5.54027665110838292360432912020, −5.33347952508187230389018035284, −4.49949305210607705725943373637, −3.87268047144181138348227829215, −2.17817477400250889627807111324, −1.38903947734257388208216929950,
1.38903947734257388208216929950, 2.17817477400250889627807111324, 3.87268047144181138348227829215, 4.49949305210607705725943373637, 5.33347952508187230389018035284, 5.54027665110838292360432912020, 6.32636697898091148536738662752, 7.33078417007987865061146672293, 7.66523349965098119975395461269, 9.520489481500963167553112831610
