L(s) = 1 | + 1.51·2-s − 3-s + 0.295·4-s − 1.86·5-s − 1.51·6-s + 1.57·7-s − 2.58·8-s + 9-s − 2.82·10-s + 0.419·11-s − 0.295·12-s + 4.28·13-s + 2.38·14-s + 1.86·15-s − 4.50·16-s + 1.70·17-s + 1.51·18-s − 5.83·19-s − 0.551·20-s − 1.57·21-s + 0.636·22-s + 23-s + 2.58·24-s − 1.51·25-s + 6.49·26-s − 27-s + 0.464·28-s + ⋯ |
L(s) = 1 | + 1.07·2-s − 0.577·3-s + 0.147·4-s − 0.834·5-s − 0.618·6-s + 0.593·7-s − 0.912·8-s + 0.333·9-s − 0.894·10-s + 0.126·11-s − 0.0853·12-s + 1.18·13-s + 0.636·14-s + 0.481·15-s − 1.12·16-s + 0.413·17-s + 0.357·18-s − 1.33·19-s − 0.123·20-s − 0.342·21-s + 0.135·22-s + 0.208·23-s + 0.527·24-s − 0.303·25-s + 1.27·26-s − 0.192·27-s + 0.0877·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 1.51T + 2T^{2} \) |
| 5 | \( 1 + 1.86T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 - 0.419T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 31 | \( 1 + 2.45T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 3.89T + 83T^{2} \) |
| 89 | \( 1 + 6.42T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−8.481671813607352521115324527151, −8.145102562087394868048100581441, −6.86842490218401650254584003369, −6.26164353872457605442089670578, −5.38115552431414949648989777807, −4.65866204721430688681813672214, −3.93202900147511801802400254663, −3.26971223588676075656076109747, −1.66180895046833995294914728141, 0,
1.66180895046833995294914728141, 3.26971223588676075656076109747, 3.93202900147511801802400254663, 4.65866204721430688681813672214, 5.38115552431414949648989777807, 6.26164353872457605442089670578, 6.86842490218401650254584003369, 8.145102562087394868048100581441, 8.481671813607352521115324527151