| 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
−16.35572748764816143780475636002, −15.75465887118622916832970225804, −13.86554479007193199005948350584, −13.10994081888771609658566724847, −11.76800058294023756470577669323, −10.51890554822255421224875049676, −8.879050035743400477216255077649, −7.07059162794068927055955639049, −5.56973455801038757937021007617, −4.28187908495290383818163061851,
3.12577587895058400483385361762, 5.73222117792192607638296650903, 6.45460612731784572822488885061, 8.824305371445970115991481892443, 10.75189390466300422569152363310, 11.28736966497198411350684047583, 12.68340616084920348294678304466, 13.64045378128344928664944308822, 15.14460572104743792398669752480, 16.01975510952175567378610134683
