L(s) = 1 | + (1.19 − 1.60i)2-s + (−1.86 − 1.07i)3-s + (−1.13 − 3.83i)4-s + (3.25 + 5.63i)5-s + (−3.96 + 1.70i)6-s + (2.39 + 6.57i)7-s + (−7.50 − 2.76i)8-s + (−2.17 − 3.76i)9-s + (12.9 + 1.52i)10-s + (−0.528 − 0.305i)11-s + (−2.01 + 8.38i)12-s − 10.6·13-s + (13.4 + 4.02i)14-s − 14.0i·15-s + (−13.4 + 8.71i)16-s + (5.99 − 10.3i)17-s + ⋯ |
L(s) = 1 | + (0.598 − 0.801i)2-s + (−0.622 − 0.359i)3-s + (−0.284 − 0.958i)4-s + (0.650 + 1.12i)5-s + (−0.660 + 0.283i)6-s + (0.342 + 0.939i)7-s + (−0.938 − 0.345i)8-s + (−0.241 − 0.418i)9-s + (1.29 + 0.152i)10-s + (−0.0480 − 0.0277i)11-s + (−0.167 + 0.699i)12-s − 0.820·13-s + (0.957 + 0.287i)14-s − 0.935i·15-s + (−0.838 + 0.544i)16-s + (0.352 − 0.610i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.961738 - 0.523283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.961738 - 0.523283i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 + 1.60i)T \) |
| 7 | \( 1 + (-2.39 - 6.57i)T \) |
good | 3 | \( 1 + (1.86 + 1.07i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-3.25 - 5.63i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (0.528 + 0.305i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 10.6T + 169T^{2} \) |
| 17 | \( 1 + (-5.99 + 10.3i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (10.5 - 6.10i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-34.7 + 20.0i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 9.04T + 841T^{2} \) |
| 31 | \( 1 + (30.2 + 17.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-25.4 - 44.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 25.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 19.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-40.7 + 23.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-13.4 + 23.2i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (39.8 + 23.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (21.1 + 36.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-24.0 - 13.8i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 57.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (28.1 - 48.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-15.9 + 9.19i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 37.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-12.3 - 21.3i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 109.T + 9.40e3T^{2} \) |
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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
−17.20091412504267350249826152248, −14.99092087930229121921047168867, −14.46228199640345065256911722043, −12.83860537581552110621678902305, −11.77052236243200125098021501065, −10.75588145909543120886002167786, −9.337226413794549213702993290903, −6.62554897490320417107988604327, −5.40680002699864825858586071695, −2.64828301866322266445469852119,
4.58637172313302428525116826240, 5.59142114806708472563221911416, 7.53452713057067378281772367836, 9.156268756148422081714435909194, 10.92863986904341272502056368984, 12.58099028297368970940328524286, 13.51426697767290520631400632803, 14.76922620959158010665993993959, 16.34628304304116516830336268176, 17.02452717205798764278261494335