| L(s) = 1 | − 2.70·2-s − 3-s + 5.31·4-s + 0.910·5-s + 2.70·6-s − 2.94·7-s − 8.95·8-s + 9-s − 2.46·10-s + 1.96·11-s − 5.31·12-s − 6.90·13-s + 7.95·14-s − 0.910·15-s + 13.5·16-s − 3.43·17-s − 2.70·18-s − 4.84·19-s + 4.83·20-s + 2.94·21-s − 5.32·22-s − 23-s + 8.95·24-s − 4.17·25-s + 18.6·26-s − 27-s − 15.6·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.577·3-s + 2.65·4-s + 0.407·5-s + 1.10·6-s − 1.11·7-s − 3.16·8-s + 0.333·9-s − 0.778·10-s + 0.593·11-s − 1.53·12-s − 1.91·13-s + 2.12·14-s − 0.235·15-s + 3.39·16-s − 0.833·17-s − 0.637·18-s − 1.11·19-s + 1.08·20-s + 0.641·21-s − 1.13·22-s − 0.208·23-s + 1.82·24-s − 0.834·25-s + 3.66·26-s − 0.192·27-s − 2.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2539558573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2539558573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 - 0.910T + 5T^{2} \) |
| 7 | \( 1 + 2.94T + 7T^{2} \) |
| 11 | \( 1 - 1.96T + 11T^{2} \) |
| 13 | \( 1 + 6.90T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 31 | \( 1 + 6.10T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 0.108T + 41T^{2} \) |
| 43 | \( 1 - 4.09T + 43T^{2} \) |
| 47 | \( 1 + 2.94T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 4.18T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 + 7.02T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−9.408785119562597256300925253618, −8.591723009707353039160916306729, −7.56979567442351276393486355695, −6.92336520402116689451457832866, −6.39451550043922061648216060924, −5.62440073200907490152196716574, −4.15062349492276183198850685706, −2.64767232545286983067214018913, −1.99152872815009521438847312757, −0.42955634591553531340071045824,
0.42955634591553531340071045824, 1.99152872815009521438847312757, 2.64767232545286983067214018913, 4.15062349492276183198850685706, 5.62440073200907490152196716574, 6.39451550043922061648216060924, 6.92336520402116689451457832866, 7.56979567442351276393486355695, 8.591723009707353039160916306729, 9.408785119562597256300925253618
