| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 4i·7-s + 2.82·8-s − 16.9i·11-s − 8i·13-s − 5.65i·14-s + 4.00·16-s − 12.7·17-s + 16·19-s − 24i·22-s + 16.9·23-s − 11.3i·26-s − 8.00i·28-s + 4.24i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.571i·7-s + 0.353·8-s − 1.54i·11-s − 0.615i·13-s − 0.404i·14-s + 0.250·16-s − 0.748·17-s + 0.842·19-s − 1.09i·22-s + 0.737·23-s − 0.435i·26-s − 0.285i·28-s + 0.146i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.559249781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559249781\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 49T^{2} \) |
| 11 | \( 1 + 16.9iT - 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 + 12.7T + 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 - 16.9T + 529T^{2} \) |
| 29 | \( 1 - 4.24iT - 841T^{2} \) |
| 31 | \( 1 - 44T + 961T^{2} \) |
| 37 | \( 1 + 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 84.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 - 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 76T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176iT - 9.40e3T^{2} \) |
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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
−10.98227499698656330687740486293, −10.03911775681122053983448495740, −8.811032152046908682329387180156, −7.902625527689662828808478376585, −6.85708493572049847456650163811, −5.91285292069588676524349662025, −4.96066667490500883761695891675, −3.73010292308083831304997130458, −2.80847192933609458864705218805, −0.887296236992764387828458722532,
1.78062869172661924812980214033, 2.95119353443851788897824654717, 4.43100603770608651664440440118, 5.06323189190742668070903336352, 6.40395852160544498554277950039, 7.08719197475371221233377542196, 8.222961778580255333538752901795, 9.390982866838975194978286840224, 10.08851724843839726043903431333, 11.35891645531985118184119722083
