L(s) = 1 | + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−3.81 − 3.22i)5-s + 2.44·6-s + (3.99 − 3.99i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−7.04 + 0.590i)10-s − 16.7·11-s + (2.44 − 2.44i)12-s + (−7.41 − 7.41i)13-s − 7.98i·14-s + (−0.723 − 8.62i)15-s − 4·16-s + (−17.8 + 17.8i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.763 − 0.645i)5-s + 0.408·6-s + (0.570 − 0.570i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.704 + 0.0590i)10-s − 1.51·11-s + (0.204 − 0.204i)12-s + (−0.570 − 0.570i)13-s − 0.570i·14-s + (−0.0482 − 0.575i)15-s − 0.250·16-s + (−1.05 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4291307007\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4291307007\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (3.81 + 3.22i)T \) |
| 23 | \( 1 + (-3.39 - 3.39i)T \) |
good | 7 | \( 1 + (-3.99 + 3.99i)T - 49iT^{2} \) |
| 11 | \( 1 + 16.7T + 121T^{2} \) |
| 13 | \( 1 + (7.41 + 7.41i)T + 169iT^{2} \) |
| 17 | \( 1 + (17.8 - 17.8i)T - 289iT^{2} \) |
| 19 | \( 1 - 18.0iT - 361T^{2} \) |
| 29 | \( 1 + 17.7iT - 841T^{2} \) |
| 31 | \( 1 - 2.15T + 961T^{2} \) |
| 37 | \( 1 + (-34.1 + 34.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 55.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (25.9 + 25.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (1.36 - 1.36i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (17.0 + 17.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 72.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 67.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-29.6 + 29.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 21.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (73.0 + 73.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 86.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (105. + 105. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 86.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (42.2 - 42.2i)T - 9.40e3iT^{2} \) |
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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
−10.10979490877962465488733754160, −8.839237261639950774290316762014, −8.024324813953008033106038707942, −7.47063514369303918162857867167, −5.81345676193364746209092794036, −4.83951499157911674048413457410, −4.23393185757953166940005740224, −3.17310528410218447389197787654, −1.86092673422857230281718239594, −0.11226075244447698180644956035,
2.38875379639431380932255339781, 3.00174734812399550577118371183, 4.55794970741028643591788689547, 5.15883082414138207838648573126, 6.63958482690766659405187127157, 7.17272118092165870733775222075, 8.050022550130990500450882571713, 8.660242915646263198946324645845, 9.817064372328351220271411231162, 11.11900064495871754808607043056