| L(s) = 1 | − 45.2·2-s − 39.0i·3-s + 2.04e3·4-s − 2.86e3i·5-s + 1.76e3i·6-s − 9.26e4·8-s + 5.29e5·9-s + 1.29e5i·10-s + 3.41e6·11-s − 7.98e4i·12-s + 6.97e6i·13-s − 1.11e5·15-s + 4.19e6·16-s − 2.85e7i·17-s − 2.39e7·18-s + 5.25e7i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0534i·3-s + 0.500·4-s − 0.183i·5-s + 0.0378i·6-s − 0.353·8-s + 0.997·9-s + 0.129i·10-s + 1.92·11-s − 0.0267i·12-s + 1.44i·13-s − 0.00981·15-s + 0.250·16-s − 1.18i·17-s − 0.705·18-s + 1.11i·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
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.996350429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.996350429\) |
| \(L(7)\) |
|
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 | \( 1 + 39.0iT - 5.31e5T^{2} \) |
| 5 | \( 1 + 2.86e3iT - 2.44e8T^{2} \) |
| 11 | \( 1 - 3.41e6T + 3.13e12T^{2} \) |
| 13 | \( 1 - 6.97e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + 2.85e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 5.25e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 - 3.11e7T + 2.19e16T^{2} \) |
| 29 | \( 1 - 4.56e8T + 3.53e17T^{2} \) |
| 31 | \( 1 - 1.75e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 1.81e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 5.14e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 7.27e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + 4.28e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 7.22e9T + 4.91e20T^{2} \) |
| 59 | \( 1 + 5.15e9iT - 1.77e21T^{2} \) |
| 61 | \( 1 + 5.11e9iT - 2.65e21T^{2} \) |
| 67 | \( 1 + 7.59e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + 1.13e11T + 1.64e22T^{2} \) |
| 73 | \( 1 - 1.74e11iT - 2.29e22T^{2} \) |
| 79 | \( 1 - 2.26e11T + 5.90e22T^{2} \) |
| 83 | \( 1 + 4.13e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 2.60e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 - 4.10e11iT - 6.93e23T^{2} \) |
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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
−11.79485243484478067408020840794, −10.40360272937795844805096486806, −9.349420916342490718535864219416, −8.718497027318496962484964493870, −7.03682124347567899295331804997, −6.63322615739397426333429352511, −4.74258391043739892551798460603, −3.56800057497212342662445352824, −1.72914623875469020006621837105, −1.08631312697660743851870308712,
0.68539640001291376286124492967, 1.59664393543216727209536992051, 3.18290856888910913650746212514, 4.43733779399124218621599332415, 6.16991298354388249139983625556, 7.02748446873887717895634555791, 8.247019329910789854320401658783, 9.319053925125599229387370680939, 10.24518889715317823469012914725, 11.21182649071656712748691329099
