| L(s) = 1 | + 0.477·3-s + 4.18·5-s − 2.77·9-s + 2·11-s + 4.66·13-s + 2·15-s − 3.23·17-s + 0.954·19-s − 23-s + 12.5·25-s − 2.75·27-s − 5.77·29-s + 7.89·31-s + 0.954·33-s − 2·37-s + 2.22·39-s + 4.66·41-s − 4·43-s − 11.6·45-s + 6.94·47-s − 1.54·51-s − 6·53-s + 8.37·55-s + 0.455·57-s + 6.09·59-s + 10.6·61-s + 19.5·65-s + ⋯ |
| L(s) = 1 | + 0.275·3-s + 1.87·5-s − 0.924·9-s + 0.603·11-s + 1.29·13-s + 0.516·15-s − 0.784·17-s + 0.219·19-s − 0.208·23-s + 2.50·25-s − 0.530·27-s − 1.07·29-s + 1.41·31-s + 0.166·33-s − 0.328·37-s + 0.356·39-s + 0.728·41-s − 0.609·43-s − 1.73·45-s + 1.01·47-s − 0.216·51-s − 0.824·53-s + 1.12·55-s + 0.0603·57-s + 0.793·59-s + 1.36·61-s + 2.42·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.642027292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.642027292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.477T + 3T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 - 0.954T + 19T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 5.14T + 89T^{2} \) |
| 97 | \( 1 + 1.32T + 97T^{2} \) |
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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
−7.88857774372504814991670155555, −6.65423370982427679747218690135, −6.41956222600459456484794510211, −5.71230744357026704575466891026, −5.24318830736628588357801418259, −4.18755794857485421440663591021, −3.32207933308834506502468833350, −2.48465869241063483023589735497, −1.86118873650837656149226189546, −0.945257234792380444361515515689,
0.945257234792380444361515515689, 1.86118873650837656149226189546, 2.48465869241063483023589735497, 3.32207933308834506502468833350, 4.18755794857485421440663591021, 5.24318830736628588357801418259, 5.71230744357026704575466891026, 6.41956222600459456484794510211, 6.65423370982427679747218690135, 7.88857774372504814991670155555
