| L(s) = 1 | + (0.483 + 1.32i)2-s + (−1.53 + 1.28i)4-s + (4.76 − 5.68i)5-s + (−1.37 − 2.38i)7-s + (−2.44 − 1.41i)8-s + (9.85 + 3.58i)10-s + (−11.5 − 6.66i)11-s + (−3.42 − 19.4i)13-s + (2.50 − 2.98i)14-s + (0.694 − 3.93i)16-s + (3.53 + 9.71i)17-s + (−13.2 − 13.5i)19-s + 14.8i·20-s + (3.27 − 18.5i)22-s + (5.22 + 6.22i)23-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (−0.383 + 0.321i)4-s + (0.953 − 1.13i)5-s + (−0.196 − 0.340i)7-s + (−0.306 − 0.176i)8-s + (0.985 + 0.358i)10-s + (−1.04 − 0.605i)11-s + (−0.263 − 1.49i)13-s + (0.178 − 0.213i)14-s + (0.0434 − 0.246i)16-s + (0.207 + 0.571i)17-s + (−0.698 − 0.715i)19-s + 0.741i·20-s + (0.148 − 0.843i)22-s + (0.227 + 0.270i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42146 - 0.790799i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42146 - 0.790799i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.483 - 1.32i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (13.2 + 13.5i)T \) |
| good | 5 | \( 1 + (-4.76 + 5.68i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (1.37 + 2.38i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (11.5 + 6.66i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (3.42 + 19.4i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-3.53 - 9.71i)T + (-221. + 185. i)T^{2} \) |
| 23 | \( 1 + (-5.22 - 6.22i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (5.76 - 15.8i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (21.6 + 37.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 59.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-48.0 - 8.47i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-34.4 - 28.9i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-9.43 + 25.9i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-21.6 - 25.7i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-30.6 - 84.1i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-5.98 + 5.02i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (73.2 + 26.6i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-22.5 + 26.8i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (16.4 - 93.0i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-15.4 + 87.7i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (62.6 - 36.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (161. - 28.4i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (16.8 - 6.12i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−11.00973264249472684644782413495, −10.10600592393429349506622278238, −9.167682669376204659779188165657, −8.270015366983460098542489822157, −7.45510437151597006427699208346, −5.85187814211357310369000606583, −5.53871262320286554618636567504, −4.33783806151484677501358624129, −2.70373121783457868018949048217, −0.67672904093450249127748793791,
2.05320862751649681105579123380, 2.73917752804920757491656725765, 4.29099967519152712548362034173, 5.56775352209523669213965585824, 6.50924095005648541357386614533, 7.50994035775707236800179609294, 9.071150650327036844359076690311, 9.800983661837716948075879342165, 10.54245554233831164008278178304, 11.31574173460045864869503732791
