| L(s) = 1 | + (−1.17 − 1.61i)2-s + (−1.23 + 3.80i)4-s − 8.50·7-s + (7.60 − 2.47i)8-s − 1.79i·11-s + 0.472i·13-s + (10 + 13.7i)14-s + (−12.9 − 9.40i)16-s − 23.8i·17-s + 9.40i·19-s + (−2.90 + 2.11i)22-s − 16.1·23-s + (0.763 − 0.555i)26-s + (10.5 − 32.3i)28-s + 6.94·29-s + ⋯ |
| L(s) = 1 | + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s − 1.21·7-s + (0.951 − 0.309i)8-s − 0.163i·11-s + 0.0363i·13-s + (0.714 + 0.983i)14-s + (−0.809 − 0.587i)16-s − 1.40i·17-s + 0.494i·19-s + (−0.132 + 0.0959i)22-s − 0.700·23-s + (0.0293 − 0.0213i)26-s + (0.375 − 1.15i)28-s + 0.239·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8055496411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8055496411\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 + 1.61i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8.50T + 49T^{2} \) |
| 11 | \( 1 + 1.79iT - 121T^{2} \) |
| 13 | \( 1 - 0.472iT - 169T^{2} \) |
| 17 | \( 1 + 23.8iT - 289T^{2} \) |
| 19 | \( 1 - 9.40iT - 361T^{2} \) |
| 23 | \( 1 + 16.1T + 529T^{2} \) |
| 29 | \( 1 - 6.94T + 841T^{2} \) |
| 31 | \( 1 - 47.4iT - 961T^{2} \) |
| 37 | \( 1 + 26.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.00T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 21.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 88.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 21.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 39.1iT - 9.40e3T^{2} \) |
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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
−9.880728161399999288963229348639, −9.304252671027426134070280685352, −8.513652310590105657070156980545, −7.44754342899708586345789149229, −6.74190422027930041119489528069, −5.54560626605655653331468059600, −4.25296613671668522621888590801, −3.28526335484262312061227001212, −2.44899298569836810492336082234, −0.847770143156409966253232548996,
0.44305415883779527692579486907, 2.08383745098272626407257036812, 3.61385669398416320054661965834, 4.68365978142767998362163555586, 6.05177402714949374746172830416, 6.28189971337260335031249718124, 7.39087445711923100960652242706, 8.159071992234400530976423906596, 9.081513049986280866916353570435, 9.770436452911257587198111364560
