| L(s) = 1 | + 1.73i·3-s − 2.23i·5-s + (−2.44 − 6.55i)7-s − 2.99·9-s − 14.4·11-s − 16.9i·13-s + 3.87·15-s − 13.0i·17-s + 18.6i·19-s + (11.3 − 4.23i)21-s − 10.3·23-s − 5.00·25-s − 5.19i·27-s − 13.7·29-s + 42.4i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + (−0.349 − 0.936i)7-s − 0.333·9-s − 1.31·11-s − 1.30i·13-s + 0.258·15-s − 0.765i·17-s + 0.983i·19-s + (0.540 − 0.201i)21-s − 0.451·23-s − 0.200·25-s − 0.192i·27-s − 0.473·29-s + 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5726627123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5726627123\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (2.44 + 6.55i)T \) |
| good | 11 | \( 1 + 14.4T + 121T^{2} \) |
| 13 | \( 1 + 16.9iT - 169T^{2} \) |
| 17 | \( 1 + 13.0iT - 289T^{2} \) |
| 19 | \( 1 - 18.6iT - 361T^{2} \) |
| 23 | \( 1 + 10.3T + 529T^{2} \) |
| 29 | \( 1 + 13.7T + 841T^{2} \) |
| 31 | \( 1 - 42.4iT - 961T^{2} \) |
| 37 | \( 1 - 28.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 5.84T + 1.84e3T^{2} \) |
| 47 | \( 1 - 10.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 81.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 35.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 68.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 47.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 47.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 125. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 129.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 42.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 25.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 28.7iT - 9.40e3T^{2} \) |
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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
−9.594787228145161763293753052422, −8.550871122770266789368755808907, −7.84465852857280311485536155550, −7.24478730276835452322372957078, −5.93511925882131465711887721865, −5.30474910340241453219631774134, −4.46529161385500833634146577620, −3.48920805037342699661073763888, −2.66038322466872895884676131703, −0.971404846410572419043636044939,
0.17345569999265498074969917233, 2.05285083507307950809495239590, 2.51919289934668495543740566020, 3.73527378652012348319698321250, 4.91403024879496221911592688311, 5.86762568950603588315313518617, 6.44546498896854529984797745613, 7.35464313675429303075691519514, 8.054986471595120802223406496950, 8.936741696878624114453593867830
