L(s) = 1 | + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−3.67 − 2.12i)5-s + 2.82i·8-s + (−3 − 5.19i)10-s + (−14.6 + 8.48i)11-s + 8·13-s + (−2.00 + 3.46i)16-s + (11.0 − 6.36i)17-s + (8 − 13.8i)19-s − 8.48i·20-s − 24·22-s + (−14.6 − 8.48i)23-s + (−3.5 − 6.06i)25-s + (9.79 + 5.65i)26-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.734 − 0.424i)5-s + 0.353i·8-s + (−0.300 − 0.519i)10-s + (−1.33 + 0.771i)11-s + 0.615·13-s + (−0.125 + 0.216i)16-s + (0.648 − 0.374i)17-s + (0.421 − 0.729i)19-s − 0.424i·20-s − 1.09·22-s + (−0.638 − 0.368i)23-s + (−0.140 − 0.242i)25-s + (0.376 + 0.217i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9729485140\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9729485140\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (3.67 + 2.12i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (14.6 - 8.48i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 8T + 169T^{2} \) |
| 17 | \( 1 + (-11.0 + 6.36i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-8 + 13.8i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (14.6 + 8.48i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 + (22 + 38.1i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-17 + 29.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + (73.4 + 42.4i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (33.0 - 19.0i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (29.3 - 16.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25 - 43.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-8 - 13.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38 + 65.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-11.0 - 6.36i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 176T + 9.40e3T^{2} \) |
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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
−9.718861287618962810862312970447, −8.625743716045049690397585013642, −7.77481368250929035814868093605, −7.34376044999115391717451069356, −6.08827357587359628402551617166, −5.17620082455211142542757654344, −4.42394723127628673032919476038, −3.41923875675352544471554605053, −2.20146259006655249048725598261, −0.25185809599347690812056891714,
1.50739936844273005844238514381, 3.16231261535100966756269436107, 3.49185881912232052415658736361, 4.86261662107052543545178495424, 5.70977104975185188391608674850, 6.58454296442981762452584461267, 7.88319340296108836626617634738, 8.115905302315093308519249158592, 9.575788108422536757903021822449, 10.42335135301303332799310485745