| L(s) = 1 | + 45.2·2-s − 1.27e3i·3-s + 2.04e3·4-s + 2.86e4i·5-s − 5.78e4i·6-s + 9.26e4·8-s − 1.10e6·9-s + 1.29e6i·10-s + 6.36e5·11-s − 2.61e6i·12-s − 3.67e6i·13-s + 3.65e7·15-s + 4.19e6·16-s − 1.29e7i·17-s − 4.98e7·18-s + 2.20e7i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75i·3-s + 0.500·4-s + 1.83i·5-s − 1.23i·6-s + 0.353·8-s − 2.07·9-s + 1.29i·10-s + 0.359·11-s − 0.876i·12-s − 0.761i·13-s + 3.21·15-s + 0.250·16-s − 0.537i·17-s − 1.46·18-s + 0.468i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.04845357828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04845357828\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 45.2T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.27e3iT - 5.31e5T^{2} \) |
| 5 | \( 1 - 2.86e4iT - 2.44e8T^{2} \) |
| 11 | \( 1 - 6.36e5T + 3.13e12T^{2} \) |
| 13 | \( 1 + 3.67e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + 1.29e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 2.20e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + 1.09e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + 1.71e8T + 3.53e17T^{2} \) |
| 31 | \( 1 - 1.40e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 6.20e8T + 6.58e18T^{2} \) |
| 41 | \( 1 + 2.48e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 1.21e10T + 3.99e19T^{2} \) |
| 47 | \( 1 + 6.14e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 3.72e10T + 4.91e20T^{2} \) |
| 59 | \( 1 - 5.83e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 + 3.95e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 + 1.48e10T + 8.18e21T^{2} \) |
| 71 | \( 1 - 3.43e10T + 1.64e22T^{2} \) |
| 73 | \( 1 - 2.46e11iT - 2.29e22T^{2} \) |
| 79 | \( 1 + 3.43e11T + 5.90e22T^{2} \) |
| 83 | \( 1 - 3.79e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 5.83e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 5.78e11iT - 6.93e23T^{2} \) |
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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
−11.30646487082832204020610485788, −10.17203348492472962323760712233, −8.087521133975675825818475033914, −7.21369472087961789795569303065, −6.56152790149466040668982590082, −5.70088429473155010712267367339, −3.45800756033556783403800509340, −2.60850915970142229924350642631, −1.63169600075278565194829555021, −0.00743063310814564424262964325,
1.67564681573292623029467103361, 3.53620212742487772926296550385, 4.45634109730987039826082087446, 4.94882705231401700556318817058, 6.06693762049530058698068815200, 8.245293864666684007837562738351, 9.145132722836713459490295537086, 9.901505387736072439088252918049, 11.25815105180188584290360062139, 12.09280379437606074849890130385
