| L(s) = 1 | + (−1.99 − 0.000176i)2-s + (1.35 + 1.07i)3-s + (3.99 + 0.000706i)4-s + (−1.20 − 0.578i)5-s + (−2.70 − 2.16i)6-s + (1.99 + 1.59i)7-s + (−7.99 − 0.00211i)8-s + (0.667 + 2.92i)9-s + (2.40 + 1.15i)10-s + (−17.9 − 4.09i)11-s + (5.41 + 4.32i)12-s + (−1.50 + 6.61i)13-s + (−3.99 − 3.19i)14-s + (−1.00 − 2.08i)15-s + (15.9 + 0.00565i)16-s − 6.77·17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 8.83e−5i)2-s + (0.451 + 0.359i)3-s + (0.999 + 0.000176i)4-s + (−0.240 − 0.115i)5-s + (−0.451 − 0.360i)6-s + (0.285 + 0.227i)7-s + (−0.999 − 0.000264i)8-s + (0.0741 + 0.324i)9-s + (0.240 + 0.115i)10-s + (−1.62 − 0.371i)11-s + (0.451 + 0.360i)12-s + (−0.116 + 0.508i)13-s + (−0.285 − 0.227i)14-s + (−0.0668 − 0.138i)15-s + (0.999 + 0.000353i)16-s − 0.398·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0335200 + 0.315722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0335200 + 0.315722i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 + 0.000176i)T \) |
| 3 | \( 1 + (-1.35 - 1.07i)T \) |
| 29 | \( 1 + (-28.9 - 2.39i)T \) |
| good | 5 | \( 1 + (1.20 + 0.578i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (-1.99 - 1.59i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (17.9 + 4.09i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (1.50 - 6.61i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + 6.77T + 289T^{2} \) |
| 19 | \( 1 + (15.2 - 12.1i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (-4.38 - 9.10i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (12.3 - 25.6i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (4.70 + 20.5i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 + 48.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-23.6 - 49.1i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (69.6 + 15.8i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (31.8 + 15.3i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 - 83.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (34.6 - 43.4i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (-10.5 + 2.41i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (117. + 26.8i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-13.3 + 6.45i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (44.5 - 10.1i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (8.60 - 6.86i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-128. - 61.8i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (11.0 + 13.9i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.43032440255769708650937803635, −10.54072878979128458373268673969, −9.941722364579348280063829423369, −8.688313956687374990743443425302, −8.262104465603910023852845502345, −7.31792895124153008297732325846, −6.03327306458407324499055691297, −4.76484221901808720124095094634, −3.15436733555015467305169395913, −1.98352838334630805482843797178,
0.17285784696541035738616278081, 2.03852727129264835241211017336, 3.09423211642373246665059417649, 4.90105627971825692677128010048, 6.33612991183376933672457132696, 7.42365985219521575263082754830, 7.977364847114946142669249332247, 8.817760405692494828416203400121, 9.978750961999580359871851481269, 10.67153992940171735397661952280
