| L(s) = 1 | − 5.83·5-s + 5.24·7-s − 1.24·11-s − 5.89i·13-s + 1.57·17-s + (10.6 + 15.7i)19-s − 27.5·23-s + 9.05·25-s − 15.9i·29-s + 53.2i·31-s − 30.5·35-s − 10.0i·37-s + 69.8i·41-s − 52.9·43-s − 12.2·47-s + ⋯ |
| L(s) = 1 | − 1.16·5-s + 0.748·7-s − 0.112·11-s − 0.453i·13-s + 0.0923·17-s + (0.561 + 0.827i)19-s − 1.19·23-s + 0.362·25-s − 0.548i·29-s + 1.71i·31-s − 0.873·35-s − 0.270i·37-s + 1.70i·41-s − 1.23·43-s − 0.259·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6977853732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6977853732\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-10.6 - 15.7i)T \) |
| good | 5 | \( 1 + 5.83T + 25T^{2} \) |
| 7 | \( 1 - 5.24T + 49T^{2} \) |
| 11 | \( 1 + 1.24T + 121T^{2} \) |
| 13 | \( 1 + 5.89iT - 169T^{2} \) |
| 17 | \( 1 - 1.57T + 289T^{2} \) |
| 23 | \( 1 + 27.5T + 529T^{2} \) |
| 29 | \( 1 + 15.9iT - 841T^{2} \) |
| 31 | \( 1 - 53.2iT - 961T^{2} \) |
| 37 | \( 1 + 10.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 69.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 52.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 40.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 75.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 47.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 56.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 42.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 24.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.19iT - 9.40e3T^{2} \) |
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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
−10.59401080489905338694694567519, −9.858228130360184958078987231212, −8.533010750554832880573658311201, −7.993564052932961005359462810772, −7.34805409298520164189879623608, −6.08532707740946755374925652087, −4.99976748799280892968627292648, −4.08187386070969521769062774930, −3.08318722441429235623511473807, −1.42815468977774913979078584803,
0.25838410733803230837064089618, 1.96498909845106040170784204427, 3.47193090738914745815172225881, 4.36365489735586528380207373482, 5.26360138611430058653923691395, 6.54610648075803267417499257512, 7.58875360194467756001390886472, 8.056848307398555025523398206119, 9.014098467833217488373642441458, 10.00608479488041967020348651657
