| L(s) = 1 | + (0.892 + 0.157i)2-s + (−1.71 + 0.245i)3-s + (−1.10 − 0.403i)4-s + (−1.14 − 0.415i)5-s + (−1.56 − 0.0508i)6-s + (−1.98 − 1.74i)7-s + (−2.49 − 1.44i)8-s + (2.87 − 0.841i)9-s + (−0.954 − 0.550i)10-s + (−1.22 − 3.37i)11-s + (1.99 + 0.419i)12-s + (−0.206 + 0.566i)13-s + (−1.49 − 1.87i)14-s + (2.06 + 0.432i)15-s + (−0.194 − 0.163i)16-s + (−3.97 + 6.88i)17-s + ⋯ |
| L(s) = 1 | + (0.631 + 0.111i)2-s + (−0.989 + 0.141i)3-s + (−0.553 − 0.201i)4-s + (−0.510 − 0.185i)5-s + (−0.640 − 0.0207i)6-s + (−0.750 − 0.661i)7-s + (−0.882 − 0.509i)8-s + (0.959 − 0.280i)9-s + (−0.301 − 0.174i)10-s + (−0.369 − 1.01i)11-s + (0.576 + 0.121i)12-s + (−0.0571 + 0.157i)13-s + (−0.399 − 0.500i)14-s + (0.532 + 0.111i)15-s + (−0.0485 − 0.0407i)16-s + (−0.964 + 1.66i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.142886 - 0.367989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.142886 - 0.367989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 - 0.245i)T \) |
| 7 | \( 1 + (1.98 + 1.74i)T \) |
| good | 2 | \( 1 + (-0.892 - 0.157i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (1.14 + 0.415i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (1.22 + 3.37i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.206 - 0.566i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.97 - 6.88i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.22 + 0.707i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.00 + 0.177i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.413 + 1.13i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.83 + 7.79i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 8.45T + 37T^{2} \) |
| 41 | \( 1 + (6.73 + 2.45i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.41 + 8.02i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.296 - 0.107i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (5.74 - 3.31i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.763 - 0.640i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (4.19 + 11.5i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.79 + 10.1i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.952 - 0.550i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 0.933iT - 73T^{2} \) |
| 79 | \( 1 + (2.33 - 13.2i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.66 + 2.06i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (2.96 + 5.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.29 - 1.46i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.39037847628785626041798705350, −11.25845742309647080374914583237, −10.38764532765808998945495924827, −9.370191525900461293377773176350, −8.056316167219245197630245984761, −6.49880514916265745987054591282, −5.84880255371010384387344770408, −4.46169152641622966445418229049, −3.71501102131295439300615263927, −0.32142130068366690055472440277,
2.90688377210575592446001959894, 4.49788728117668515664326487983, 5.27766476242506743094176513445, 6.55602027039046719832192212751, 7.61051881757611226379319740226, 9.145511465228176018683359632329, 9.994532631137838896162181058025, 11.45406748061828450581390541901, 12.00432191647041742904762437830, 12.86797472731325100990840828891
