| L(s) = 1 | − 1.41·2-s + 2.00·4-s + (4.25 + 2.61i)5-s + 4.80i·7-s − 2.82·8-s + (−6.02 − 3.70i)10-s + 4.90i·11-s − 14.9i·13-s − 6.79i·14-s + 4.00·16-s + 6.09·17-s + 14.5·19-s + (8.51 + 5.23i)20-s − 6.93i·22-s + 42.3·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + (0.851 + 0.523i)5-s + 0.686i·7-s − 0.353·8-s + (−0.602 − 0.370i)10-s + 0.445i·11-s − 1.14i·13-s − 0.485i·14-s + 0.250·16-s + 0.358·17-s + 0.764·19-s + (0.425 + 0.261i)20-s − 0.315i·22-s + 1.84·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.598353462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598353462\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.25 - 2.61i)T \) |
| good | 7 | \( 1 - 4.80iT - 49T^{2} \) |
| 11 | \( 1 - 4.90iT - 121T^{2} \) |
| 13 | \( 1 + 14.9iT - 169T^{2} \) |
| 17 | \( 1 - 6.09T + 289T^{2} \) |
| 19 | \( 1 - 14.5T + 361T^{2} \) |
| 23 | \( 1 - 42.3T + 529T^{2} \) |
| 29 | \( 1 - 40.0iT - 841T^{2} \) |
| 31 | \( 1 + 43.1T + 961T^{2} \) |
| 37 | \( 1 + 49.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 4.72T + 2.20e3T^{2} \) |
| 53 | \( 1 - 19.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 75.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.65T + 3.72e3T^{2} \) |
| 67 | \( 1 - 59.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 73.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 124.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 83.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 138. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 10.0iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21779857893770353022774571119, −9.227128850419806970465276478282, −8.815009339088175639281658615064, −7.46733053123899708936517192909, −6.99616730232757412094022737268, −5.67933771500275232625728949803, −5.28931656540780073913533984255, −3.31687761610192837937589818267, −2.50336476477363904080042140253, −1.21316781173789608804599935891,
0.78733397063894561451241145209, 1.83320329725644845109360000895, 3.22236421733177379605793395276, 4.57126951809446886353285347722, 5.58809648284728838101908003768, 6.58597616396258552395641473005, 7.34016954467567979801834910562, 8.369686822736254982198679197499, 9.265586744401869455307722350888, 9.654483462323154401860467476087
