| L(s) = 1 | + (−1.95 − 0.344i)3-s + (−6.26 − 5.26i)5-s + (−0.733 − 1.27i)7-s + (−4.76 − 1.73i)9-s + (4.00 − 6.94i)11-s + (3.08 − 0.544i)13-s + (10.4 + 12.4i)15-s + (−24.4 + 8.89i)17-s + (18.9 + 0.657i)19-s + (0.994 + 2.73i)21-s + (15.0 − 12.5i)23-s + (7.29 + 41.3i)25-s + (24.1 + 13.9i)27-s + (14.4 − 39.7i)29-s + (−23.5 + 13.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.650 − 0.114i)3-s + (−1.25 − 1.05i)5-s + (−0.104 − 0.181i)7-s + (−0.529 − 0.192i)9-s + (0.364 − 0.630i)11-s + (0.237 − 0.0418i)13-s + (0.695 + 0.828i)15-s + (−1.43 + 0.523i)17-s + (0.999 + 0.0346i)19-s + (0.0473 + 0.130i)21-s + (0.652 − 0.547i)23-s + (0.291 + 1.65i)25-s + (0.894 + 0.516i)27-s + (0.499 − 1.37i)29-s + (−0.759 + 0.438i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.200603 - 0.485312i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.200603 - 0.485312i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-18.9 - 0.657i)T \) |
| good | 3 | \( 1 + (1.95 + 0.344i)T + (8.45 + 3.07i)T^{2} \) |
| 5 | \( 1 + (6.26 + 5.26i)T + (4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (0.733 + 1.27i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.00 + 6.94i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-3.08 + 0.544i)T + (158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (24.4 - 8.89i)T + (221. - 185. i)T^{2} \) |
| 23 | \( 1 + (-15.0 + 12.5i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-14.4 + 39.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (23.5 - 13.5i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 34.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (41.3 + 7.29i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (25.7 + 21.5i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-13.2 - 4.81i)T + (1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (11.9 + 14.1i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (38.5 + 106. i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-5.25 + 4.41i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (32.1 - 88.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-40.4 + 48.2i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (11.1 - 63.1i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-131. - 23.2i)T + (5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-64.9 - 112. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-27.7 + 4.89i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-28.3 - 78.0i)T + (-7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.68273659365561520312764220337, −12.53868294136272600739429560090, −11.67019875340776020364288410771, −10.95164358027521267161000297007, −9.028991377481961721691521916216, −8.217821815419482528014159915368, −6.66609365794603773410581280494, −5.17771336435345805109962272489, −3.79097842901238620748879468387, −0.48344068088433366788816730537,
3.15241882530247940228863612709, 4.80288293442029925251078458198, 6.53810412772634464736304235887, 7.49152899879044370034717581442, 8.997844939756619749191902790042, 10.63413386662833916336900580677, 11.40264583791568737213249593698, 12.06477688240140977525384509120, 13.68060198976209157468388130561, 14.90102189132641763628584586503
