L(s) = 1 | − 1.56i·2-s + 3i·3-s + 5.56·4-s + 4.68·6-s + 10.2i·7-s − 21.1i·8-s − 9·9-s − 11·11-s + 16.6i·12-s − 40.8i·13-s + 16·14-s + 11.4·16-s + 98.7i·17-s + 14.0i·18-s + 39.6·19-s + ⋯ |
L(s) = 1 | − 0.552i·2-s + 0.577i·3-s + 0.695·4-s + 0.318·6-s + 0.553i·7-s − 0.935i·8-s − 0.333·9-s − 0.301·11-s + 0.401i·12-s − 0.872i·13-s + 0.305·14-s + 0.178·16-s + 1.40i·17-s + 0.184i·18-s + 0.478·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.466827841\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.466827841\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 + 1.56iT - 8T^{2} \) |
| 7 | \( 1 - 10.2iT - 343T^{2} \) |
| 13 | \( 1 + 40.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 98.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 39.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 61.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 44.8iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 2.36iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 333. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 640. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 370.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 714.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 404. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 939.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 362. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 951.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 735. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 966. iT - 9.12e5T^{2} \) |
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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
−10.13915588432836945707031121517, −9.257705698762947169675105758507, −8.240813991951058354199027709656, −7.44905908278291343809159284209, −6.16963853289000229626982528889, −5.60775682028971814147111514000, −4.29112399073061607463657642798, −3.23792624014631241278749856679, −2.45178039961771338546466069177, −1.09779961072852877610754903335,
0.72278701715807835849541370458, 2.08914800236058875932364256952, 3.04546132905551729018468224439, 4.55848591655620503068830589161, 5.53424029963534709263242785939, 6.61170701674766123987470769893, 7.06657404974051795656999885010, 7.81741707568331076485590608876, 8.689433553891857824650466115657, 9.756634208335109515918847414332