L(s) = 1 | + (−0.425 + 1.34i)2-s + (−1.61 + 1.84i)3-s + (−1.63 − 1.14i)4-s + (−0.00363 + 0.0554i)5-s + (−1.79 − 2.96i)6-s + (2.12 − 1.57i)7-s + (2.24 − 1.71i)8-s + (−0.391 − 2.97i)9-s + (−0.0732 − 0.0285i)10-s + (1.96 − 5.78i)11-s + (4.75 − 1.16i)12-s + (2.24 − 1.50i)13-s + (1.22 + 3.53i)14-s + (−0.0962 − 0.0962i)15-s + (1.36 + 3.76i)16-s + (−0.0852 − 0.318i)17-s + ⋯ |
L(s) = 1 | + (−0.301 + 0.953i)2-s + (−0.932 + 1.06i)3-s + (−0.818 − 0.574i)4-s + (−0.00162 + 0.0248i)5-s + (−0.733 − 1.20i)6-s + (0.802 − 0.596i)7-s + (0.794 − 0.607i)8-s + (−0.130 − 0.991i)9-s + (−0.0231 − 0.00901i)10-s + (0.592 − 1.74i)11-s + (1.37 − 0.334i)12-s + (0.622 − 0.416i)13-s + (0.326 + 0.945i)14-s + (−0.0248 − 0.0248i)15-s + (0.340 + 0.940i)16-s + (−0.0206 − 0.0771i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.787503 + 0.271882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.787503 + 0.271882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.425 - 1.34i)T \) |
| 7 | \( 1 + (-2.12 + 1.57i)T \) |
good | 3 | \( 1 + (1.61 - 1.84i)T + (-0.391 - 2.97i)T^{2} \) |
| 5 | \( 1 + (0.00363 - 0.0554i)T + (-4.95 - 0.652i)T^{2} \) |
| 11 | \( 1 + (-1.96 + 5.78i)T + (-8.72 - 6.69i)T^{2} \) |
| 13 | \( 1 + (-2.24 + 1.50i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (0.0852 + 0.318i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (6.14 + 3.03i)T + (11.5 + 15.0i)T^{2} \) |
| 23 | \( 1 + (-0.893 - 6.78i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-1.81 + 9.10i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (4.72 + 2.72i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0177 + 0.271i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-9.05 - 3.74i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (0.291 + 1.46i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-0.342 + 1.27i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-8.74 - 2.96i)T + (42.0 + 32.2i)T^{2} \) |
| 59 | \( 1 + (-4.42 - 8.98i)T + (-35.9 + 46.8i)T^{2} \) |
| 61 | \( 1 + (9.59 - 3.25i)T + (48.3 - 37.1i)T^{2} \) |
| 67 | \( 1 + (6.31 + 5.53i)T + (8.74 + 66.4i)T^{2} \) |
| 71 | \( 1 + (2.36 + 5.71i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.58 - 4.67i)T + (-18.8 + 70.5i)T^{2} \) |
| 79 | \( 1 + (-1.11 + 4.17i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-7.38 + 4.93i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-0.958 - 0.735i)T + (23.0 + 85.9i)T^{2} \) |
| 97 | \( 1 + 8.72T + 97T^{2} \) |
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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
−10.95105728565762813540788758959, −10.48132089688222464703901664698, −9.255546462187233009786914472526, −8.554110960287474930888759402611, −7.53510553542079669773426665447, −6.18985444654146786330682313215, −5.68823356548461144215983469645, −4.57160285583982860276970501676, −3.80335908690754027522904357899, −0.75657423045446353955667419616,
1.39429466115159623646529258098, 2.19368484070746312152008001367, 4.21456684220929311406097325544, 5.10300927557249394356609142369, 6.46866227315031169948205208840, 7.26973713092530371787660199957, 8.452491653351249937681791870221, 9.134051661702951682714888749667, 10.54957529254590190783366541694, 11.01063723387262022374000421591