L(s) = 1 | − 1.68i·2-s + (−1.02 + 1.39i)3-s − 0.855·4-s − 1.45·5-s + (2.35 + 1.73i)6-s + (0.366 − 2.62i)7-s − 1.93i·8-s + (−0.881 − 2.86i)9-s + 2.46i·10-s − i·11-s + (0.880 − 1.19i)12-s − 3.51i·13-s + (−4.42 − 0.619i)14-s + (1.50 − 2.03i)15-s − 4.97·16-s + 2.75·17-s + ⋯ |
L(s) = 1 | − 1.19i·2-s + (−0.594 + 0.804i)3-s − 0.427·4-s − 0.652·5-s + (0.961 + 0.710i)6-s + (0.138 − 0.990i)7-s − 0.683i·8-s + (−0.293 − 0.955i)9-s + 0.779i·10-s − 0.301i·11-s + (0.254 − 0.344i)12-s − 0.975i·13-s + (−1.18 − 0.165i)14-s + (0.387 − 0.524i)15-s − 1.24·16-s + 0.667·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.331946 - 0.790227i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.331946 - 0.790227i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.02 - 1.39i)T \) |
| 7 | \( 1 + (-0.366 + 2.62i)T \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + 1.68iT - 2T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 13 | \( 1 + 3.51iT - 13T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 19 | \( 1 + 4.07iT - 19T^{2} \) |
| 23 | \( 1 - 4.35iT - 23T^{2} \) |
| 29 | \( 1 - 1.86iT - 29T^{2} \) |
| 31 | \( 1 - 5.81iT - 31T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 - 0.863T + 41T^{2} \) |
| 43 | \( 1 - 2.11T + 43T^{2} \) |
| 47 | \( 1 - 0.483T + 47T^{2} \) |
| 53 | \( 1 + 13.8iT - 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 + 5.37iT - 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 10.0iT - 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 8.75T + 79T^{2} \) |
| 83 | \( 1 - 7.13T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 11.7iT - 97T^{2} \) |
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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
−11.48307043484562768036882131660, −11.02475432692142259534946737616, −10.22014919568723341376929799874, −9.453260221422644138474003749594, −7.954507469389832357133863401196, −6.75595747706655052123211172031, −5.20483571374980233985557184052, −3.96689481534076318257945080986, −3.21623256525530726824122967185, −0.77270924487916296217126715457,
2.21305093397576547567865311663, 4.53969384765850083224770769132, 5.79432194875517472041682705031, 6.38921287651888351701166383469, 7.63530266773777433344853458524, 8.055876184662623308462609689936, 9.293681076614500446807158984302, 10.93629146108403157073441657522, 11.88637787208278174084419319613, 12.28030990549715048134820611914