| L(s) = 1 | + (0.939 − 0.342i)2-s + (1.33 + 1.10i)3-s + (0.766 − 0.642i)4-s + (−1.29 + 1.53i)5-s + (1.63 + 0.583i)6-s + (2.56 + 0.660i)7-s + (0.500 − 0.866i)8-s + (0.553 + 2.94i)9-s + (−0.687 + 1.88i)10-s − 2.67i·11-s + (1.73 − 0.00950i)12-s + (−1.33 + 3.65i)13-s + (2.63 − 0.255i)14-s + (−3.42 + 0.623i)15-s + (0.173 − 0.984i)16-s + (−0.293 − 0.0517i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.769 + 0.638i)3-s + (0.383 − 0.321i)4-s + (−0.577 + 0.688i)5-s + (0.665 + 0.238i)6-s + (0.968 + 0.249i)7-s + (0.176 − 0.306i)8-s + (0.184 + 0.982i)9-s + (−0.217 + 0.597i)10-s − 0.807i·11-s + (0.499 − 0.00274i)12-s + (−0.369 + 1.01i)13-s + (0.703 − 0.0682i)14-s + (−0.884 + 0.160i)15-s + (0.0434 − 0.246i)16-s + (−0.0711 − 0.0125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.63596 + 1.18727i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63596 + 1.18727i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-1.33 - 1.10i)T \) |
| 7 | \( 1 + (-2.56 - 0.660i)T \) |
| 19 | \( 1 + (-1.41 - 4.12i)T \) |
| good | 5 | \( 1 + (1.29 - 1.53i)T + (-0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + 2.67iT - 11T^{2} \) |
| 13 | \( 1 + (1.33 - 3.65i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.293 + 0.0517i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.13 + 5.87i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.13 + 2.63i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (8.01 + 4.62i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.07 - 2.35i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.17 - 0.429i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.132 + 0.751i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.0609 + 0.0107i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (1.86 - 1.56i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.95 + 11.0i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-12.7 - 4.65i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.168 + 0.462i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.31 + 13.1i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (10.0 + 8.40i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (8.33 + 1.46i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (7.99 - 4.61i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (10.5 - 8.81i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-12.3 + 14.6i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−10.59641632884398303210158080477, −9.607376394955623207537763513467, −8.636283430610280231664387219934, −7.87948656089342826314117311527, −7.00951159133557256344126578501, −5.77464962534313677426674510859, −4.70237730320802651147953586968, −3.95904670100894665922744003831, −3.00559584900683041929858016707, −1.95155397869158486315455363144,
1.22136800185677043836793690739, 2.57666024146740226097873199521, 3.76641724706529539119965738785, 4.74444765794343264150038917930, 5.49792616346303535318311413645, 7.11229266297239703156472298338, 7.40485750027077527654533476756, 8.298311014045724796852654307069, 8.970382641062954959232217876487, 10.12926331030073886592099308163
