| L(s) = 1 | + (−1.80 + 0.854i)2-s + (−1.35 − 1.07i)3-s + (2.54 − 3.08i)4-s + (5.99 + 2.88i)5-s + (3.37 + 0.796i)6-s + (−4.67 − 3.73i)7-s + (−1.95 + 7.75i)8-s + (0.667 + 2.92i)9-s + (−13.3 − 0.0994i)10-s + (10.4 + 2.38i)11-s + (−6.77 + 1.44i)12-s + (−4.68 + 20.5i)13-s + (11.6 + 2.75i)14-s + (−4.99 − 10.3i)15-s + (−3.09 − 15.6i)16-s − 12.4·17-s + ⋯ |
| L(s) = 1 | + (−0.904 + 0.427i)2-s + (−0.451 − 0.359i)3-s + (0.635 − 0.772i)4-s + (1.19 + 0.577i)5-s + (0.561 + 0.132i)6-s + (−0.668 − 0.533i)7-s + (−0.244 + 0.969i)8-s + (0.0741 + 0.324i)9-s + (−1.33 − 0.00994i)10-s + (0.950 + 0.216i)11-s + (−0.564 + 0.120i)12-s + (−0.360 + 1.58i)13-s + (0.832 + 0.196i)14-s + (−0.333 − 0.692i)15-s + (−0.193 − 0.981i)16-s − 0.729·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.922236 + 0.517122i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.922236 + 0.517122i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.80 - 0.854i)T \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
| 29 | \( 1 + (-27.9 + 7.68i)T \) |
| good | 5 | \( 1 + (-5.99 - 2.88i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (4.67 + 3.73i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-10.4 - 2.38i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (4.68 - 20.5i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + 12.4T + 289T^{2} \) |
| 19 | \( 1 + (-19.9 + 15.8i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (3.95 + 8.22i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (19.3 - 40.1i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (-13.1 - 57.7i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 - 42.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-32.4 - 67.3i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (11.6 + 2.66i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-75.4 - 36.3i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 - 55.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (-23.5 + 29.5i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (-5.70 + 1.30i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (80.4 + 18.3i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-26.0 + 12.5i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (56.6 - 12.9i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (-75.0 + 59.8i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-150. - 72.3i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (55.1 + 69.1i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.26040464394343995705766698861, −10.27147550936722682133162649824, −9.562603402394328847096457619087, −8.933715970603511677321353763367, −7.24542234274171886827174704745, −6.65388557546436441933502550022, −6.19031345542408539033717450303, −4.68836655478985936037770008520, −2.56093248384574714607779918234, −1.28582088616934593819077131923,
0.77978508628445727323785869533, 2.34530445928315877287803898400, 3.72365603684105933465078210469, 5.54580038341929642885626314478, 6.09424217253813322929200870188, 7.44409270471325672901548957852, 8.760773164558057971544345028064, 9.458529428403184286115536677814, 9.976894479210780934383250246620, 10.90634427225743477894325318687
