| L(s) = 1 | − 1.41i·2-s + 3.53i·3-s − 2.00·4-s − 2.23i·5-s + 5.00·6-s + 6.36i·7-s + 2.82i·8-s − 3.52·9-s − 3.16·10-s + 3.70i·11-s − 7.07i·12-s + 2.62i·13-s + 8.99·14-s + 7.91·15-s + 4.00·16-s − 20.8·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.17i·3-s − 0.500·4-s − 0.447i·5-s + 0.834·6-s + 0.909i·7-s + 0.353i·8-s − 0.391·9-s − 0.316·10-s + 0.336i·11-s − 0.589i·12-s + 0.202i·13-s + 0.642·14-s + 0.527·15-s + 0.250·16-s − 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8618800750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8618800750\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + 2.23iT \) |
| 67 | \( 1 + (49.5 + 45.0i)T \) |
| good | 3 | \( 1 - 3.53iT - 9T^{2} \) |
| 7 | \( 1 - 6.36iT - 49T^{2} \) |
| 11 | \( 1 - 3.70iT - 121T^{2} \) |
| 13 | \( 1 - 2.62iT - 169T^{2} \) |
| 17 | \( 1 + 20.8T + 289T^{2} \) |
| 19 | \( 1 - 4.54T + 361T^{2} \) |
| 23 | \( 1 - 20.8T + 529T^{2} \) |
| 29 | \( 1 + 36.8T + 841T^{2} \) |
| 31 | \( 1 + 4.44iT - 961T^{2} \) |
| 37 | \( 1 + 39.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 32.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 66.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 81.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 32.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 23.6iT - 3.72e3T^{2} \) |
| 71 | \( 1 + 44.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 25.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 50.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 51.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 108.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 140. iT - 9.40e3T^{2} \) |
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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
−10.69955589353586669154237373973, −9.655606698402381344841577206642, −9.157725468419181695988744638819, −8.578311725004988495197189304465, −7.17475297296641791821781034791, −5.75270767006866453918274042892, −4.88752747101178954275804431088, −4.19983918124244435044725913251, −3.05892958447119461657694069087, −1.78796552093615991405981142549,
0.30483455507029389502053596858, 1.71576498639477449556731762325, 3.27743320899159713770472669127, 4.50212718419212963614750324047, 5.74603376728331257490479507004, 6.84737356736890025574999643273, 7.04167336750913559729418270183, 7.956679963103345178723031224929, 8.822615763980662534780055660837, 9.940117638398571127453252096866
