| L(s) = 1 | − 2·2-s + (1 − 2.82i)3-s + 4·4-s + (−2 + 5.65i)6-s − 8·8-s + (−7.00 − 5.65i)9-s + (−7 − 8.48i)11-s + (4 − 11.3i)12-s + 16·16-s − 2·17-s + (14.0 + 11.3i)18-s − 16.9i·19-s + (14 + 16.9i)22-s + (−8 + 22.6i)24-s − 25·25-s + ⋯ |
| L(s) = 1 | − 2-s + (0.333 − 0.942i)3-s + 4-s + (−0.333 + 0.942i)6-s − 8-s + (−0.777 − 0.628i)9-s + (−0.636 − 0.771i)11-s + (0.333 − 0.942i)12-s + 16-s − 0.117·17-s + (0.777 + 0.628i)18-s − 0.893i·19-s + (0.636 + 0.771i)22-s + (−0.333 + 0.942i)24-s − 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.114329 - 0.646743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.114329 - 0.646743i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + (-1 + 2.82i)T \) |
| 11 | \( 1 + (7 + 8.48i)T \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 2T + 289T^{2} \) |
| 19 | \( 1 + 16.9iT - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 46T + 1.68e3T^{2} \) |
| 43 | \( 1 + 84.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 62T + 4.48e3T^{2} \) |
| 71 | \( 1 + 5.04e3T^{2} \) |
| 73 | \( 1 + 33.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 158T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−11.32627065978106592864861146579, −10.32865806456957640401268573534, −9.132922022417845522915308446752, −8.391440902035262872083499578145, −7.54175650456623042283414451699, −6.62536571140026943070190178960, −5.57392090263471256081235437610, −3.26241689648122793973265438821, −2.05531474876567140283238055154, −0.41410663730054621594562737847,
2.09435769287856637016964126806, 3.47699813429912053767108011948, 4.98341732910305405642762075367, 6.25786266993057920231026535832, 7.68845976713507922485247517972, 8.321587562292204631537915704994, 9.552767115905468611488245600374, 9.987606708717199599693871560417, 10.91779064561977793638239571376, 11.80336749437518193559562541059
