| 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 | \( 1 + (-28.3 + 78.0i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.90102189132641763628584586503, −13.68060198976209157468388130561, −12.06477688240140977525384509120, −11.40264583791568737213249593698, −10.63413386662833916336900580677, −8.997844939756619749191902790042, −7.49152899879044370034717581442, −6.53810412772634464736304235887, −4.80288293442029925251078458198, −3.15241882530247940228863612709,
0.48344068088433366788816730537, 3.79097842901238620748879468387, 5.17771336435345805109962272489, 6.66609365794603773410581280494, 8.217821815419482528014159915368, 9.028991377481961721691521916216, 10.95164358027521267161000297007, 11.67019875340776020364288410771, 12.53868294136272600739429560090, 13.68273659365561520312764220337
