| L(s) = 1 | + (−0.514 − 1.31i)2-s + (0.0980 − 0.995i)3-s + (−1.47 + 1.35i)4-s + (2.10 − 1.12i)5-s + (−1.36 + 0.382i)6-s + (−0.203 − 1.02i)7-s + (2.54 + 1.24i)8-s + (−0.980 − 0.195i)9-s + (−2.55 − 2.19i)10-s + (1.79 − 2.19i)11-s + (1.20 + 1.59i)12-s + (0.643 − 1.20i)13-s + (−1.24 + 0.792i)14-s + (−0.911 − 2.20i)15-s + (0.331 − 3.98i)16-s + (−1.09 + 2.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.363 − 0.931i)2-s + (0.0565 − 0.574i)3-s + (−0.735 + 0.677i)4-s + (0.939 − 0.502i)5-s + (−0.555 + 0.156i)6-s + (−0.0767 − 0.386i)7-s + (0.898 + 0.439i)8-s + (−0.326 − 0.0650i)9-s + (−0.809 − 0.692i)10-s + (0.542 − 0.661i)11-s + (0.347 + 0.461i)12-s + (0.178 − 0.333i)13-s + (−0.331 + 0.211i)14-s + (−0.235 − 0.568i)15-s + (0.0828 − 0.996i)16-s + (−0.264 + 0.638i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.405372 - 1.12903i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.405372 - 1.12903i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.514 + 1.31i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (-2.10 + 1.12i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.203 + 1.02i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.79 + 2.19i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.643 + 1.20i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (1.09 - 2.63i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.72 + 5.69i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.26 + 0.844i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (2.28 - 1.87i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (0.614 - 0.614i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.893 + 0.270i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-2.76 - 4.14i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.302 + 3.07i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (11.9 + 4.96i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-9.20 - 7.55i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-5.50 - 10.3i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.331 + 0.0326i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.353 - 0.0348i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-4.59 + 0.913i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.835 + 4.20i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.943 + 0.390i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-15.6 + 4.73i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-2.26 - 1.51i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-9.37 + 9.37i)T - 97iT^{2} \) |
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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
−10.97343846122797267517219182196, −10.15440925960030549664673740100, −9.023059546668778835236340269316, −8.653319107572405049137320047737, −7.35364787423446400955210331706, −6.18721386382558392099089044364, −4.98278167804093464565763638981, −3.60439452649130825013977487069, −2.22741164143963281873464587440, −0.967641383178150296102240483952,
2.01942403076201971597165048971, 3.89422522037164783072831683749, 5.12252097000342296001030909764, 6.07422300886074687700912181939, 6.80781592655488732542794135956, 8.020287976865861877647468313829, 9.120857416295881438520025823298, 9.697728568833798165624805612522, 10.36377232382254740057535847780, 11.47986350836981763379268250621
