L(s) = 1 | + (−3.44 + 3.88i)3-s + (−0.950 − 1.64i)5-s + (14.2 + 11.7i)7-s + (−3.23 − 26.8i)9-s + (33.6 + 19.4i)11-s + (−8.40 − 4.85i)13-s + (9.67 + 1.97i)15-s + (22.9 + 39.6i)17-s + (19.7 + 11.4i)19-s + (−95.0 + 14.9i)21-s + (−135. + 78.4i)23-s + (60.6 − 105. i)25-s + (115. + 79.8i)27-s + (−187. + 108. i)29-s + 201. i·31-s + ⋯ |
L(s) = 1 | + (−0.663 + 0.748i)3-s + (−0.0849 − 0.147i)5-s + (0.771 + 0.635i)7-s + (−0.119 − 0.992i)9-s + (0.923 + 0.533i)11-s + (−0.179 − 0.103i)13-s + (0.166 + 0.0340i)15-s + (0.326 + 0.565i)17-s + (0.239 + 0.138i)19-s + (−0.987 + 0.155i)21-s + (−1.23 + 0.711i)23-s + (0.485 − 0.841i)25-s + (0.822 + 0.568i)27-s + (−1.19 + 0.692i)29-s + 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.246135137\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246135137\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.44 - 3.88i)T \) |
| 7 | \( 1 + (-14.2 - 11.7i)T \) |
good | 5 | \( 1 + (0.950 + 1.64i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-33.6 - 19.4i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (8.40 + 4.85i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-22.9 - 39.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-19.7 - 11.4i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (135. - 78.4i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (187. - 108. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 201. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (146. - 253. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (119. - 206. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (158. + 275. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (26.3 - 15.1i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + 466.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 506. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 630.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 413. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-276. + 159. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-415. - 719. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (572. - 990. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.55e3 + 899. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.11795172616094059592480943800, −10.98953549674400449697099492454, −10.05240833966008462262776717664, −9.139750721848385756523104356626, −8.175757791396139954078343237780, −6.75748912230023285427708548027, −5.62867332373555221512485890503, −4.73640497888129770102907403279, −3.58971228169605403641698259255, −1.56841457956488543098678936231,
0.56468015000349558988444206191, 1.94608294189378776506603109158, 3.88058296780488399041959130717, 5.18099938077577441609612805962, 6.28288018086843651305451516496, 7.31254407564874189852904689118, 8.039782546307781137529710064826, 9.364906953502681175878210503622, 10.64393605463145812310512227640, 11.40807070613993717225363912117