L(s) = 1 | − 0.713·3-s − 5-s − 1.26·7-s − 2.49·9-s + 1.29·11-s − 3.04·13-s + 0.713·15-s − 2.63·17-s − 4.73·19-s + 0.905·21-s − 6.57·23-s + 25-s + 3.91·27-s − 5.77·29-s − 6.97·31-s − 0.923·33-s + 1.26·35-s + 3.93·37-s + 2.17·39-s − 5.22·41-s + 2.20·43-s + 2.49·45-s − 11.7·47-s − 5.38·49-s + 1.88·51-s + 8.06·53-s − 1.29·55-s + ⋯ |
L(s) = 1 | − 0.412·3-s − 0.447·5-s − 0.479·7-s − 0.830·9-s + 0.389·11-s − 0.844·13-s + 0.184·15-s − 0.639·17-s − 1.08·19-s + 0.197·21-s − 1.37·23-s + 0.200·25-s + 0.754·27-s − 1.07·29-s − 1.25·31-s − 0.160·33-s + 0.214·35-s + 0.647·37-s + 0.348·39-s − 0.816·41-s + 0.335·43-s + 0.371·45-s − 1.71·47-s − 0.769·49-s + 0.263·51-s + 1.10·53-s − 0.174·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2861965105\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2861965105\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 0.713T + 3T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 + 5.22T + 41T^{2} \) |
| 43 | \( 1 - 2.20T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 7.63T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.86T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.012996575684560818963025028486, −6.86255077045657566675423096262, −6.62689787496881774719017966632, −5.70740454048826516621119405834, −5.16080702981831929280027517891, −4.17194909877232808759408448489, −3.67111307582272838333254423575, −2.62126012039706508289503721604, −1.87306469603824933308865680751, −0.25299333943491157146991555986,
0.25299333943491157146991555986, 1.87306469603824933308865680751, 2.62126012039706508289503721604, 3.67111307582272838333254423575, 4.17194909877232808759408448489, 5.16080702981831929280027517891, 5.70740454048826516621119405834, 6.62689787496881774719017966632, 6.86255077045657566675423096262, 8.012996575684560818963025028486