L(s) = 1 | + (0.785 − 1.35i)2-s + (3.89 + 3.43i)3-s + (2.76 + 4.79i)4-s + (7.73 − 2.60i)6-s + (17.1 − 29.6i)7-s + 21.2·8-s + (3.40 + 26.7i)9-s + (13.4 − 23.2i)11-s + (−5.67 + 28.1i)12-s + (−9.74 − 16.8i)13-s + (−26.8 − 46.5i)14-s + (−5.45 + 9.44i)16-s − 29.1·17-s + (39.0 + 16.3i)18-s + 47.1·19-s + ⋯ |
L(s) = 1 | + (0.277 − 0.480i)2-s + (0.750 + 0.660i)3-s + (0.345 + 0.599i)4-s + (0.526 − 0.177i)6-s + (0.924 − 1.60i)7-s + 0.939·8-s + (0.126 + 0.991i)9-s + (0.368 − 0.638i)11-s + (−0.136 + 0.678i)12-s + (−0.207 − 0.360i)13-s + (−0.513 − 0.888i)14-s + (−0.0852 + 0.147i)16-s − 0.415·17-s + (0.511 + 0.214i)18-s + 0.568·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0478i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.28794 - 0.0787768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.28794 - 0.0787768i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.89 - 3.43i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.785 + 1.35i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-17.1 + 29.6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-13.4 + 23.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (9.74 + 16.8i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 29.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-56.6 - 98.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (40.6 - 70.3i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-5.41 - 9.38i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-221. - 384. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-169. + 294. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-118. + 204. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 609.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (7.77 + 13.4i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-108. - 187. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 65.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 711.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-478. + 829. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (261. - 452. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-400. + 694. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.47147980750435503719091438185, −10.91804821882763496505548122430, −10.12778860511443814640917835253, −8.773923131996189035860688882175, −7.76945786967126016929932370411, −7.14715207982555574950266412101, −5.00053962345010567023318146110, −3.99327548037635216679437267648, −3.18938249886562503496248048635, −1.51868884519345636625275681799,
1.61293851890339520914912192910, 2.50070813824748647952774964365, 4.58731308247031225013397652369, 5.70623821404216634747361116770, 6.73267782564197022212320240880, 7.69640267298027781453682334470, 8.783716638882080549746376658393, 9.531197609053629597767355032158, 11.03256020412045143659095019363, 11.98796820158666700618000495224