| L(s) = 1 | + (0.996 + 0.0871i)2-s + (0.984 + 0.173i)4-s + (2.23 + 0.107i)5-s + (−1.02 − 0.721i)7-s + (0.965 + 0.258i)8-s + (2.21 + 0.302i)10-s + (2.11 − 5.81i)11-s + (−0.354 − 4.04i)13-s + (−0.963 − 0.808i)14-s + (0.939 + 0.342i)16-s + (1.07 − 0.287i)17-s + (−4.49 + 2.59i)19-s + (2.18 + 0.493i)20-s + (2.61 − 5.60i)22-s + (2.75 + 3.93i)23-s + ⋯ |
| L(s) = 1 | + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.998 + 0.0482i)5-s + (−0.389 − 0.272i)7-s + (0.341 + 0.0915i)8-s + (0.700 + 0.0955i)10-s + (0.637 − 1.75i)11-s + (−0.0982 − 1.12i)13-s + (−0.257 − 0.216i)14-s + (0.234 + 0.0855i)16-s + (0.260 − 0.0698i)17-s + (−1.03 + 0.595i)19-s + (0.487 + 0.110i)20-s + (0.557 − 1.19i)22-s + (0.573 + 0.819i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.67839 - 0.603158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.67839 - 0.603158i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.996 - 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.107i)T \) |
| good | 7 | \( 1 + (1.02 + 0.721i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 5.81i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.354 + 4.04i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.07 + 0.287i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.49 - 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.75 - 3.93i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.572 - 0.480i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.766 - 4.34i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.29 - 4.82i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.72 + 4.43i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.20 - 11.1i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.919 - 1.31i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-0.785 + 0.785i)T - 53iT^{2} \) |
| 59 | \( 1 + (13.8 - 5.03i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.115 + 0.653i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.62 - 0.317i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (9.91 + 5.72i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.410 - 1.53i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.82 - 5.74i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.846 + 9.67i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-0.959 - 1.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.12 + 1.45i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.43381424422561000330893677947, −9.364862640331354656587015199252, −8.524576411071596597774564104164, −7.49679058216951740074131563817, −6.20626304334625182004455090403, −6.01755161462577638212985150743, −4.95894749925900713798072016720, −3.55736336603266967070415845969, −2.88427222944840242498312561372, −1.23760776599100421674533857262,
1.80911598494695531861351497426, 2.54345586285097347829071548087, 4.18109448856172348240423525159, 4.77303402099456945566373652103, 6.00587614831298197628790416450, 6.65301585438756801780343019212, 7.37156213813982132221462281858, 8.992108392570877784447148801310, 9.415724795741969806259864280481, 10.30044841917809386842106034639
