| L(s) = 1 | + 2.08·2-s − 3-s + 2.36·4-s + 0.182·5-s − 2.08·6-s + 1.65·7-s + 0.758·8-s + 9-s + 0.380·10-s − 1.24·11-s − 2.36·12-s + 4.57·13-s + 3.45·14-s − 0.182·15-s − 3.14·16-s + 5.83·17-s + 2.08·18-s + 0.930·19-s + 0.430·20-s − 1.65·21-s − 2.60·22-s − 23-s − 0.758·24-s − 4.96·25-s + 9.56·26-s − 27-s + 3.91·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.577·3-s + 1.18·4-s + 0.0814·5-s − 0.852·6-s + 0.625·7-s + 0.268·8-s + 0.333·9-s + 0.120·10-s − 0.376·11-s − 0.682·12-s + 1.27·13-s + 0.924·14-s − 0.0470·15-s − 0.785·16-s + 1.41·17-s + 0.492·18-s + 0.213·19-s + 0.0962·20-s − 0.361·21-s − 0.555·22-s − 0.208·23-s − 0.154·24-s − 0.993·25-s + 1.87·26-s − 0.192·27-s + 0.739·28-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
| \(L(1)\) |
\(\approx\) |
\(3.711178303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.711178303\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 0.182T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 0.930T + 19T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 - 2.01T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 - 6.30T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 0.787T + 53T^{2} \) |
| 59 | \( 1 - 0.995T + 59T^{2} \) |
| 61 | \( 1 + 8.77T + 61T^{2} \) |
| 67 | \( 1 - 0.890T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 0.935T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 - 0.195T + 97T^{2} \) |
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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
−9.209095129325739850076017636037, −8.122445424463389014821235729582, −7.46984480511545190324916012816, −6.28427169686620936040787997344, −5.84256744624395195967786854012, −5.18468062954644261138452624934, −4.29717501346909414197381885239, −3.60995789740797198750085780054, −2.52293701334562237988718680829, −1.15854712705736386611733094614,
1.15854712705736386611733094614, 2.52293701334562237988718680829, 3.60995789740797198750085780054, 4.29717501346909414197381885239, 5.18468062954644261138452624934, 5.84256744624395195967786854012, 6.28427169686620936040787997344, 7.46984480511545190324916012816, 8.122445424463389014821235729582, 9.209095129325739850076017636037
