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
−11.45410185291431297438655777438, −10.06422501331001102470474027728, −9.295851262618791867369434764074, −8.464479145086259953913972394557, −7.02012847328318067404837017881, −6.87385687033231355903133856827, −5.38262431481411844876865864130, −3.83201478457675731207833263551, −2.99943941520756086234579769719, −1.26249232944580282929539657365,
0.988773456321840096921911578636, 3.19965589230321246544901146498, 3.70556857378775831330688220435, 5.29763910641396879466430096734, 6.16507670423165173271854037628, 7.55499541062025969981741958631, 8.373437053177500745611451829889, 9.257776578303105961186484004217, 10.12538756297075825834762795854, 10.94850936893815201802593905444