L(s) = 1 | − 1.30·2-s + (2.38 − 1.82i)3-s − 2.29·4-s + (−3.11 + 2.38i)6-s + 2.64i·7-s + 8.22·8-s + (2.35 − 8.68i)9-s − 2.61i·11-s + (−5.45 + 4.17i)12-s − 6.35i·13-s − 3.45i·14-s − 1.58·16-s − 12.1·17-s + (−3.07 + 11.3i)18-s + 10.2·19-s + ⋯ |
L(s) = 1 | − 0.653·2-s + (0.794 − 0.607i)3-s − 0.572·4-s + (−0.519 + 0.397i)6-s + 0.377i·7-s + 1.02·8-s + (0.261 − 0.965i)9-s − 0.237i·11-s + (−0.454 + 0.348i)12-s − 0.488i·13-s − 0.247i·14-s − 0.0989·16-s − 0.714·17-s + (−0.170 + 0.630i)18-s + 0.538·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.077607346\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077607346\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.38 + 1.82i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 2 | \( 1 + 1.30T + 4T^{2} \) |
| 11 | \( 1 + 2.61iT - 121T^{2} \) |
| 13 | \( 1 + 6.35iT - 169T^{2} \) |
| 17 | \( 1 + 12.1T + 289T^{2} \) |
| 19 | \( 1 - 10.2T + 361T^{2} \) |
| 23 | \( 1 + 4.30T + 529T^{2} \) |
| 29 | \( 1 + 17.3iT - 841T^{2} \) |
| 31 | \( 1 - 39.2T + 961T^{2} \) |
| 37 | \( 1 + 41.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 30.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 105.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 41.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 20.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 67.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 63.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 89.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 63.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.1iT - 9.40e3T^{2} \) |
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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
−10.02809934037085374594927936998, −9.330823880560495332713727731586, −8.541776199069127745432346004100, −7.961863763937424386263523271877, −7.03018600041851590034532236139, −5.84131575992603927473072184239, −4.52779986446550508201064997650, −3.30896058108153887011519338116, −1.96660470912345887567370175019, −0.51384961675670253655508881327,
1.49430658549953275189948256997, 3.08157218231671675833505204568, 4.34614190532121129123610430305, 4.88828456008543954761204456845, 6.60488348089188884070645342899, 7.75826459956492602153338775815, 8.342406198486195647794591195592, 9.306385599599901091274349570808, 9.786232999456804074037865175613, 10.59674838324702636480122935214