| L(s) = 1 | + 64·2-s + 4.09e3·4-s + 1.56e4·5-s + 1.70e5·7-s + 2.62e5·8-s + 1.00e6·10-s + 6.06e6·11-s + 3.29e7·13-s + 1.08e7·14-s + 1.67e7·16-s − 5.09e7·17-s − 3.38e8·19-s + 6.40e7·20-s + 3.88e8·22-s + 9.47e8·23-s + 2.44e8·25-s + 2.10e9·26-s + 6.96e8·28-s − 5.81e9·29-s + 3.79e9·31-s + 1.07e9·32-s − 3.25e9·34-s + 2.65e9·35-s + 2.84e10·37-s − 2.16e10·38-s + 4.09e9·40-s − 2.20e10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.546·7-s + 0.353·8-s + 0.316·10-s + 1.03·11-s + 1.89·13-s + 0.386·14-s + 0.250·16-s − 0.511·17-s − 1.64·19-s + 0.223·20-s + 0.730·22-s + 1.33·23-s + 0.199·25-s + 1.33·26-s + 0.273·28-s − 1.81·29-s + 0.767·31-s + 0.176·32-s − 0.361·34-s + 0.244·35-s + 1.82·37-s − 1.16·38-s + 0.158·40-s − 0.724·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(5.015021858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.015021858\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 64T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 1.56e4T \) |
| good | 7 | \( 1 - 1.70e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 6.06e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.29e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 5.09e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.38e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 9.47e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.81e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.79e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.84e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.20e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.49e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 5.50e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.46e9T + 2.60e22T^{2} \) |
| 59 | \( 1 + 2.80e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 5.28e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.51e9T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.83e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.21e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 6.58e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.01e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.09e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 8.64e12T + 6.73e25T^{2} \) |
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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
−11.31498734075581191899484002670, −10.93375245214196358365069386690, −9.259317813070838562069830868822, −8.288699237840976414601022173810, −6.67204673541375069170574623237, −5.97184931713374338844459700821, −4.55456510521502712932410666826, −3.59592234561946669702751907118, −2.05218082830931223823705038442, −1.07150956033166232483533623629,
1.07150956033166232483533623629, 2.05218082830931223823705038442, 3.59592234561946669702751907118, 4.55456510521502712932410666826, 5.97184931713374338844459700821, 6.67204673541375069170574623237, 8.288699237840976414601022173810, 9.259317813070838562069830868822, 10.93375245214196358365069386690, 11.31498734075581191899484002670
