| L(s) = 1 | + (0.766 − 0.642i)2-s + (−1.43 − 0.524i)3-s + (0.173 − 0.984i)4-s + (0.347 + 1.96i)5-s + (−1.43 + 0.524i)6-s + (−1.34 + 2.33i)7-s + (−0.500 − 0.866i)8-s + (−0.5 − 0.419i)9-s + (1.53 + 1.28i)10-s + (−1.59 − 2.75i)11-s + (−0.766 + 1.32i)12-s + (5.41 − 1.96i)13-s + (0.467 + 2.65i)14-s + (0.532 − 3.01i)15-s + (−0.939 − 0.342i)16-s + (−4.99 + 4.18i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.831 − 0.302i)3-s + (0.0868 − 0.492i)4-s + (0.155 + 0.880i)5-s + (−0.587 + 0.213i)6-s + (−0.509 + 0.882i)7-s + (−0.176 − 0.306i)8-s + (−0.166 − 0.139i)9-s + (0.484 + 0.406i)10-s + (−0.480 − 0.831i)11-s + (−0.221 + 0.383i)12-s + (1.50 − 0.546i)13-s + (0.125 + 0.709i)14-s + (0.137 − 0.779i)15-s + (−0.234 − 0.0855i)16-s + (−1.21 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.747454 - 0.209326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.747454 - 0.209326i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-2.82 + 3.31i)T \) |
| good | 3 | \( 1 + (1.43 + 0.524i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.347 - 1.96i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.41 + 1.96i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (4.99 - 4.18i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.120 - 0.684i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.16 + 1.81i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.22 - 2.12i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 + (-0.326 - 0.118i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.05 - 5.97i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.04 - 5.06i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.42 + 8.08i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.439 + 0.368i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.509 + 2.89i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.79 + 3.18i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.46 + 8.32i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (14.8 + 5.39i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-8.51 - 3.10i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (4.23 - 7.34i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.27 - 2.64i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.266 - 0.223i)T + (16.8 - 95.5i)T^{2} \) |
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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
−16.01975510952175567378610134683, −15.14460572104743792398669752480, −13.64045378128344928664944308822, −12.68340616084920348294678304466, −11.28736966497198411350684047583, −10.75189390466300422569152363310, −8.824305371445970115991481892443, −6.45460612731784572822488885061, −5.73222117792192607638296650903, −3.12577587895058400483385361762,
4.28187908495290383818163061851, 5.56973455801038757937021007617, 7.07059162794068927055955639049, 8.879050035743400477216255077649, 10.51890554822255421224875049676, 11.76800058294023756470577669323, 13.10994081888771609658566724847, 13.86554479007193199005948350584, 15.75465887118622916832970225804, 16.35572748764816143780475636002
