L(s) = 1 | + (−1.22 − 1.22i)3-s + (1.15 + 1.37i)5-s + (−0.0374 − 0.102i)7-s + (−0.00480 + 2.99i)9-s + (1.39 + 1.16i)11-s + (0.628 + 3.56i)13-s + (0.273 − 3.09i)15-s + (3.51 − 2.03i)17-s + (−0.846 − 0.488i)19-s + (−0.0803 + 0.171i)21-s + (5.93 + 2.16i)23-s + (0.309 − 1.75i)25-s + (3.68 − 3.66i)27-s + (0.912 + 0.160i)29-s + (0.185 − 0.508i)31-s + ⋯ |
L(s) = 1 | + (−0.706 − 0.707i)3-s + (0.515 + 0.614i)5-s + (−0.0141 − 0.0389i)7-s + (−0.00160 + 0.999i)9-s + (0.419 + 0.352i)11-s + (0.174 + 0.988i)13-s + (0.0705 − 0.799i)15-s + (0.852 − 0.492i)17-s + (−0.194 − 0.112i)19-s + (−0.0175 + 0.0375i)21-s + (1.23 + 0.450i)23-s + (0.0618 − 0.350i)25-s + (0.708 − 0.705i)27-s + (0.169 + 0.0298i)29-s + (0.0332 − 0.0913i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23502 + 0.124674i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23502 + 0.124674i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-1.15 - 1.37i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (0.0374 + 0.102i)T + (-5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.39 - 1.16i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.628 - 3.56i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.51 + 2.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.846 + 0.488i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.93 - 2.16i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.912 - 0.160i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.185 + 0.508i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.79 - 8.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0246 + 0.00434i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.93 - 8.25i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.54 + 2.74i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 + (-1.98 + 1.66i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (5.80 - 2.11i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (13.5 - 2.39i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.650 - 1.12i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.34 - 5.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.71 + 1.53i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.02 - 17.1i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (4.27 + 2.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.6 + 8.91i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.39679080499808501329318742974, −10.33757895275652422707179707837, −9.562423743218919341673237224628, −8.344680740863200637012484200847, −7.10562391678807612545627211951, −6.66121567400877865938242490644, −5.63661687735718882591371264231, −4.51632024862067896016006353067, −2.81194922589134198919891223661, −1.42835552225765145950531735605,
1.04037712848531684433911299793, 3.18148748252161103523562040318, 4.39777979697522654823538568777, 5.52000305131785547783702614892, 5.99611760160292194769015732082, 7.37567000646452572115646186659, 8.728464255765027512640770044467, 9.296853802966634788386834465604, 10.40868133991711994995904777489, 10.86643352025455197608417078357