L(s) = 1 | + 3.27i·2-s + (41.2 + 22.0i)3-s + 117.·4-s − 240.·5-s + (−72.0 + 134. i)6-s − 1.07e3i·7-s + 802. i·8-s + (1.21e3 + 1.81e3i)9-s − 786. i·10-s + 6.03e3·11-s + (4.83e3 + 2.58e3i)12-s + 4.78e3·13-s + 3.52e3·14-s + (−9.90e3 − 5.29e3i)15-s + 1.23e4·16-s + 1.44e3·17-s + ⋯ |
L(s) = 1 | + 0.289i·2-s + (0.882 + 0.471i)3-s + 0.916·4-s − 0.859·5-s + (−0.136 + 0.255i)6-s − 1.18i·7-s + 0.554i·8-s + (0.556 + 0.831i)9-s − 0.248i·10-s + 1.36·11-s + (0.808 + 0.431i)12-s + 0.604·13-s + 0.343·14-s + (−0.758 − 0.404i)15-s + 0.755·16-s + 0.0715·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.97173 + 1.05467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.97173 + 1.05467i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-41.2 - 22.0i)T \) |
| 23 | \( 1 + (-2.26e4 + 5.37e4i)T \) |
good | 2 | \( 1 - 3.27iT - 128T^{2} \) |
| 5 | \( 1 + 240.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.07e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 6.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.78e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.44e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.08e4iT - 8.93e8T^{2} \) |
| 29 | \( 1 - 4.06e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.69e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.09e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 9.12e4iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 5.83e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.33e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 8.45e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.36e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 2.30e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 2.80e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 4.47e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 1.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.40e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 9.89e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.11e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.19e7iT - 8.07e13T^{2} \) |
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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
−13.79271765368356963916090575237, −12.17302632891300067581061969279, −11.06587998487386035788028978660, −10.10008413898146009716718727434, −8.507602946893241180000125684351, −7.58206905766085876136067641777, −6.52770606867408202162745468981, −4.27477340480770211104591703870, −3.34844371050820641232741326551, −1.38643179481506569181911211389,
1.26834144669233467792696013498, 2.65557036714567864051874634164, 3.82840213690391437719038054576, 6.20079308114120209036415678593, 7.29021271016490523247054897167, 8.516319310836918703474594646580, 9.505500971437599044294581730151, 11.42157691972541436292142017606, 11.87693231466012966688608533389, 12.95905960647489093742673023135