| L(s) = 1 | + 64·2-s − 1.71e3·3-s + 4.09e3·4-s + 2.50e4·5-s − 1.09e5·6-s + 2.62e5·8-s + 1.33e6·9-s + 1.60e6·10-s − 1.70e6·11-s − 7.00e6·12-s + 9.03e6·13-s − 4.29e7·15-s + 1.67e7·16-s − 1.80e8·17-s + 8.53e7·18-s − 3.40e7·19-s + 1.02e8·20-s − 1.08e8·22-s + 1.05e9·23-s − 4.48e8·24-s − 5.91e8·25-s + 5.78e8·26-s + 4.44e8·27-s + 2.18e9·29-s − 2.74e9·30-s + 6.05e8·31-s + 1.07e9·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.35·3-s + 0.5·4-s + 0.718·5-s − 0.958·6-s + 0.353·8-s + 0.836·9-s + 0.507·10-s − 0.289·11-s − 0.677·12-s + 0.518·13-s − 0.973·15-s + 0.250·16-s − 1.81·17-s + 0.591·18-s − 0.166·19-s + 0.359·20-s − 0.204·22-s + 1.48·23-s − 0.479·24-s − 0.484·25-s + 0.366·26-s + 0.221·27-s + 0.683·29-s − 0.688·30-s + 0.122·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 64T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.71e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 2.50e4T + 1.22e9T^{2} \) |
| 11 | \( 1 + 1.70e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 9.03e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.80e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.40e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.05e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.18e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.05e8T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.78e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.48e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.24e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 7.35e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.90e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.10e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 5.59e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 8.94e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 6.75e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.80e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.14e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.34e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.69e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.01e13T + 6.73e25T^{2} \) |
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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
−11.06702446255537126444996916491, −10.19800080015179733505732018671, −8.723200109663437291591596373906, −6.91006605678568760605446129525, −6.20680464832336442963840331521, −5.27256446059532719413964273617, −4.37478636534289174884230806949, −2.66968351003020280468280891809, −1.34615869643209178957658255562, 0,
1.34615869643209178957658255562, 2.66968351003020280468280891809, 4.37478636534289174884230806949, 5.27256446059532719413964273617, 6.20680464832336442963840331521, 6.91006605678568760605446129525, 8.723200109663437291591596373906, 10.19800080015179733505732018671, 11.06702446255537126444996916491
