| L(s) = 1 | + (1.91 + 0.406i)2-s + (−0.0970 + 1.72i)3-s + (1.67 + 0.744i)4-s + (2.20 − 0.343i)5-s + (−0.889 + 3.27i)6-s + (0.690 − 1.19i)7-s + (−0.268 − 0.194i)8-s + (−2.98 − 0.335i)9-s + (4.36 + 0.240i)10-s + (−2.41 − 0.513i)11-s + (−1.44 + 2.81i)12-s + (−3.14 + 0.668i)13-s + (1.80 − 2.00i)14-s + (0.380 + 3.85i)15-s + (−2.88 − 3.20i)16-s + (3.02 + 2.19i)17-s + ⋯ |
| L(s) = 1 | + (1.35 + 0.287i)2-s + (−0.0560 + 0.998i)3-s + (0.836 + 0.372i)4-s + (0.988 − 0.153i)5-s + (−0.363 + 1.33i)6-s + (0.260 − 0.451i)7-s + (−0.0948 − 0.0689i)8-s + (−0.993 − 0.111i)9-s + (1.38 + 0.0761i)10-s + (−0.728 − 0.154i)11-s + (−0.418 + 0.813i)12-s + (−0.872 + 0.185i)13-s + (0.483 − 0.536i)14-s + (0.0981 + 0.995i)15-s + (−0.721 − 0.800i)16-s + (0.734 + 0.533i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.14643 + 1.06709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14643 + 1.06709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0970 - 1.72i)T \) |
| 5 | \( 1 + (-2.20 + 0.343i)T \) |
| good | 2 | \( 1 + (-1.91 - 0.406i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.690 + 1.19i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.41 + 0.513i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.14 - 0.668i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.02 - 2.19i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.234 - 0.170i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.389 + 0.432i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.389 + 3.71i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.895 - 8.51i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.37 + 7.31i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (6.21 - 1.32i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (3.20 - 5.54i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.750 + 7.14i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-6.69 + 4.86i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.90 + 0.404i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-12.8 - 2.72i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (1.05 - 10.0i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (10.4 - 7.59i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.44 - 4.43i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.66 - 15.8i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.184 + 0.0821i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (3.30 + 10.1i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.15 + 10.9i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.65149221140935411820033726797, −11.58864033738547427254897120465, −10.35704142006206804333987796276, −9.775509690257185270976570003058, −8.518368201964769148622414886118, −6.95241528947918410697528413127, −5.64096951527383129210664694203, −5.15024958358352213503980170990, −4.05111032356920334331507121341, −2.73000832022930014587681479074,
2.11550674966413175735722515466, 2.99189197328607221152975437309, 5.07078119251077265574879878628, 5.58786806444977730864720872370, 6.69721149062030205827385064698, 7.88275020311444815097754782723, 9.196585332403709450823209442374, 10.45877441917974704725006984354, 11.68552956458789431394096923732, 12.26748937941340603770096168566
