| L(s) = 1 | + (−0.959 − 0.281i)2-s + (1.73 + 1.50i)3-s + (0.841 + 0.540i)4-s + (−0.406 + 2.82i)5-s + (−1.24 − 1.93i)6-s + (2.38 + 1.14i)7-s + (−0.654 − 0.755i)8-s + (0.326 + 2.27i)9-s + (1.18 − 2.59i)10-s + (−0.284 − 0.970i)11-s + (0.648 + 2.20i)12-s + (−3.12 − 1.42i)13-s + (−1.96 − 1.76i)14-s + (−4.96 + 4.30i)15-s + (0.415 + 0.909i)16-s + (−0.250 + 0.160i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (1.00 + 0.870i)3-s + (0.420 + 0.270i)4-s + (−0.181 + 1.26i)5-s + (−0.507 − 0.790i)6-s + (0.901 + 0.432i)7-s + (−0.231 − 0.267i)8-s + (0.108 + 0.757i)9-s + (0.375 − 0.821i)10-s + (−0.0859 − 0.292i)11-s + (0.187 + 0.637i)12-s + (−0.865 − 0.395i)13-s + (−0.525 − 0.473i)14-s + (−1.28 + 1.11i)15-s + (0.103 + 0.227i)16-s + (−0.0607 + 0.0390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.999884 + 0.885380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999884 + 0.885380i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (-2.38 - 1.14i)T \) |
| 23 | \( 1 + (3.22 + 3.55i)T \) |
| good | 3 | \( 1 + (-1.73 - 1.50i)T + (0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.406 - 2.82i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (0.284 + 0.970i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (3.12 + 1.42i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.250 - 0.160i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.14 - 1.37i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.42 - 2.84i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.54 + 3.93i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-3.13 + 0.450i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.66 - 0.239i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-8.73 - 7.57i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 7.57iT - 47T^{2} \) |
| 53 | \( 1 + (-7.98 + 3.64i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (6.61 + 3.02i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (6.57 + 7.59i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.29 + 14.6i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (1.97 + 0.581i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (1.61 - 2.51i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (15.2 + 6.95i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.06 - 7.39i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (0.142 - 0.164i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.49 + 17.3i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.44091668149723450250340827793, −10.74055092939242351297249797794, −9.931119701579905738122184274502, −9.157733550534093243307658629089, −8.067973093280524071287963945647, −7.54628103318355939785826712254, −6.08476816557775150316703114257, −4.47838660570585882839239525206, −3.17331772278950566072020124150, −2.38698331978442225831445105114,
1.18848041589197151245117606422, 2.31995350373036581514612270137, 4.29662346657366907912149731158, 5.44262607866103894060881404557, 7.23617621546592342898485197977, 7.62393471608318231566494358185, 8.551974757696910527909614634923, 9.135099380152872245516164097359, 10.22086470010256456886305033967, 11.61876726308756368324117390595
