| L(s) = 1 | + (−2.17 − 0.793i)2-s + (−1.72 − 0.102i)3-s + (2.58 + 2.17i)4-s + (0.443 − 2.51i)5-s + (3.68 + 1.59i)6-s + (0.766 − 0.642i)7-s + (−1.59 − 2.76i)8-s + (2.97 + 0.353i)9-s + (−2.95 + 5.12i)10-s + (−0.776 − 4.40i)11-s + (−4.25 − 4.01i)12-s + (−3.79 + 1.38i)13-s + (−2.17 + 0.793i)14-s + (−1.02 + 4.30i)15-s + (0.113 + 0.645i)16-s + (−3.83 + 6.64i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.560i)2-s + (−0.998 − 0.0590i)3-s + (1.29 + 1.08i)4-s + (0.198 − 1.12i)5-s + (1.50 + 0.650i)6-s + (0.289 − 0.242i)7-s + (−0.564 − 0.978i)8-s + (0.993 + 0.117i)9-s + (−0.935 + 1.62i)10-s + (−0.234 − 1.32i)11-s + (−1.22 − 1.16i)12-s + (−1.05 + 0.383i)13-s + (−0.582 + 0.211i)14-s + (−0.264 + 1.11i)15-s + (0.0284 + 0.161i)16-s + (−0.930 + 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0133391 + 0.224027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0133391 + 0.224027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.102i)T \) |
| 7 | \( 1 + (-0.766 + 0.642i)T \) |
| good | 2 | \( 1 + (2.17 + 0.793i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.443 + 2.51i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.776 + 4.40i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (3.79 - 1.38i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.83 - 6.64i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.85 + 3.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.63 + 2.21i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (4.86 + 1.77i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 0.928i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.0540 + 0.0935i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.384 - 0.139i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.74 + 9.90i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.45 + 4.57i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 0.954T + 53T^{2} \) |
| 59 | \( 1 + (1.41 - 8.00i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (7.67 - 6.44i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.16 + 1.87i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.33 - 5.78i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.45 + 7.72i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.65 + 0.603i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.72 + 0.992i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.103 - 0.178i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.561 + 3.18i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.75452294614551937581849831076, −10.87364207051234921175055610748, −10.24147036186198696566548726019, −8.997268436512900862578310811132, −8.395752170217274255775531286839, −7.14287307309200110919637628566, −5.76953619524674501352696156489, −4.42615814605219309145335069610, −1.88358419646325367211316945742, −0.35404258830266140992300998388,
2.16224904120376890891402546910, 4.81864365642231808430142858257, 6.22143512476862294687969489297, 7.16384901499740295797414877426, 7.64261086585210181446530666114, 9.440743403589320832728022308099, 9.968197880373366000721188482990, 10.80932420068936895857457581008, 11.65127210936675210767875989775, 12.79543372865808093712589263185
