L(s) = 1 | + (1.22 + 1.22i)3-s + (−1.69 − 1.69i)5-s − 5.74·7-s + 2.99i·9-s + (5.59 − 5.59i)11-s + (13.5 − 13.5i)13-s − 4.16i·15-s + 19.7·17-s + (21.6 + 21.6i)19-s + (−7.03 − 7.03i)21-s + 24.9·23-s − 19.2i·25-s + (−3.67 + 3.67i)27-s + (−1.50 + 1.50i)29-s + 2.20i·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.339 − 0.339i)5-s − 0.820·7-s + 0.333i·9-s + (0.508 − 0.508i)11-s + (1.04 − 1.04i)13-s − 0.277i·15-s + 1.15·17-s + (1.14 + 1.14i)19-s + (−0.334 − 0.334i)21-s + 1.08·23-s − 0.768i·25-s + (−0.136 + 0.136i)27-s + (−0.0519 + 0.0519i)29-s + 0.0709i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.208i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.82829 - 0.192871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82829 - 0.192871i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
good | 5 | \( 1 + (1.69 + 1.69i)T + 25iT^{2} \) |
| 7 | \( 1 + 5.74T + 49T^{2} \) |
| 11 | \( 1 + (-5.59 + 5.59i)T - 121iT^{2} \) |
| 13 | \( 1 + (-13.5 + 13.5i)T - 169iT^{2} \) |
| 17 | \( 1 - 19.7T + 289T^{2} \) |
| 19 | \( 1 + (-21.6 - 21.6i)T + 361iT^{2} \) |
| 23 | \( 1 - 24.9T + 529T^{2} \) |
| 29 | \( 1 + (1.50 - 1.50i)T - 841iT^{2} \) |
| 31 | \( 1 - 2.20iT - 961T^{2} \) |
| 37 | \( 1 + (27.6 + 27.6i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (21.4 - 21.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 76.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-56.5 - 56.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-48.0 + 48.0i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-51.5 + 51.5i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (63.4 + 63.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 43.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 73.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 4.12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (38.4 + 38.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 52.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 23.1T + 9.40e3T^{2} \) |
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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
−10.94850936893815201802593905444, −10.12538756297075825834762795854, −9.257776578303105961186484004217, −8.373437053177500745611451829889, −7.55499541062025969981741958631, −6.16507670423165173271854037628, −5.29763910641396879466430096734, −3.70556857378775831330688220435, −3.19965589230321246544901146498, −0.988773456321840096921911578636,
1.26249232944580282929539657365, 2.99943941520756086234579769719, 3.83201478457675731207833263551, 5.38262431481411844876865864130, 6.87385687033231355903133856827, 7.02012847328318067404837017881, 8.464479145086259953913972394557, 9.295851262618791867369434764074, 10.06422501331001102470474027728, 11.45410185291431297438655777438