| L(s) = 1 | + (−1.99 − 0.0646i)2-s + (1.35 + 1.07i)3-s + (3.99 + 0.258i)4-s + (1.51 + 0.727i)5-s + (−2.63 − 2.24i)6-s + (−3.11 − 2.48i)7-s + (−7.96 − 0.774i)8-s + (0.667 + 2.92i)9-s + (−2.97 − 1.55i)10-s + (14.2 + 3.26i)11-s + (5.12 + 4.66i)12-s + (0.0624 − 0.273i)13-s + (6.06 + 5.16i)14-s + (1.26 + 2.61i)15-s + (15.8 + 2.06i)16-s + 3.51·17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0323i)2-s + (0.451 + 0.359i)3-s + (0.997 + 0.0645i)4-s + (0.302 + 0.145i)5-s + (−0.439 − 0.374i)6-s + (−0.444 − 0.354i)7-s + (−0.995 − 0.0967i)8-s + (0.0741 + 0.324i)9-s + (−0.297 − 0.155i)10-s + (1.29 + 0.296i)11-s + (0.427 + 0.388i)12-s + (0.00480 − 0.0210i)13-s + (0.432 + 0.368i)14-s + (0.0840 + 0.174i)15-s + (0.991 + 0.128i)16-s + 0.207·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29860 + 0.414273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29860 + 0.414273i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 + 0.0646i)T \) |
| 3 | \( 1 + (-1.35 - 1.07i)T \) |
| 29 | \( 1 + (19.9 - 21.0i)T \) |
| good | 5 | \( 1 + (-1.51 - 0.727i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (3.11 + 2.48i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-14.2 - 3.26i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (-0.0624 + 0.273i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 - 3.51T + 289T^{2} \) |
| 19 | \( 1 + (-11.7 + 9.39i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (-18.4 - 38.3i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-21.8 + 45.2i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (-15.0 - 65.9i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 - 7.86T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-8.14 - 16.9i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (-61.9 - 14.1i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-84.2 - 40.5i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 - 6.46iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (39.7 - 49.7i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (-93.8 + 21.4i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (5.04 + 1.15i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (72.2 - 34.7i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (-20.0 + 4.58i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (31.9 - 25.4i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (153. + 74.1i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (44.5 + 55.8i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.27853717396056074212115469841, −10.04065774527102210581216268942, −9.585548986060639392754926401597, −8.847046568543341517428668613900, −7.62414004806280541665921612772, −6.86432038114720762443793851791, −5.75709953778275498948804587323, −3.99524380297362337114349478475, −2.83128362137631755718000076439, −1.27505704106529824738656902557,
1.00029978489075800397137318846, 2.38518731738073529748828379693, 3.68358604203510866323913100763, 5.70016356095793106347202088591, 6.58927680106251929212616502654, 7.46129524316049812576747579875, 8.674910647772981489790821770561, 9.140622072423795309136730877473, 9.997247380216805622722040450685, 11.09213994712396553597198309455
