| L(s) = 1 | − 2.47·2-s − 1.08·3-s + 4.10·4-s − 5-s + 2.66·6-s + 0.388·7-s − 5.20·8-s − 1.83·9-s + 2.47·10-s − 4.04·11-s − 4.43·12-s + 4.93·13-s − 0.959·14-s + 1.08·15-s + 4.65·16-s − 4.62·17-s + 4.53·18-s + 4.46·19-s − 4.10·20-s − 0.419·21-s + 10.0·22-s + 6.96·23-s + 5.62·24-s + 25-s − 12.1·26-s + 5.22·27-s + 1.59·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.623·3-s + 2.05·4-s − 0.447·5-s + 1.08·6-s + 0.146·7-s − 1.84·8-s − 0.611·9-s + 0.781·10-s − 1.22·11-s − 1.28·12-s + 1.36·13-s − 0.256·14-s + 0.278·15-s + 1.16·16-s − 1.12·17-s + 1.06·18-s + 1.02·19-s − 0.918·20-s − 0.0915·21-s + 2.13·22-s + 1.45·23-s + 1.14·24-s + 0.200·25-s − 2.39·26-s + 1.00·27-s + 0.301·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 + 1.08T + 3T^{2} \) |
| 7 | \( 1 - 0.388T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 - 1.99T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 0.688T + 43T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 9.56T + 59T^{2} \) |
| 61 | \( 1 - 0.491T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 2.07T + 71T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 - 5.20T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 0.748T + 89T^{2} \) |
| 97 | \( 1 - 2.07T + 97T^{2} \) |
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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
−8.565481036376774186760022078734, −8.351866406305791369536367915036, −7.30499549849997451792723201496, −6.76844784794317292000507173582, −5.75240330767887151932259258216, −4.98580618004634516900454670676, −3.44444435415039176083481227061, −2.42908936091757690316360734309, −1.09514482161144769982914379695, 0,
1.09514482161144769982914379695, 2.42908936091757690316360734309, 3.44444435415039176083481227061, 4.98580618004634516900454670676, 5.75240330767887151932259258216, 6.76844784794317292000507173582, 7.30499549849997451792723201496, 8.351866406305791369536367915036, 8.565481036376774186760022078734
