| L(s) = 1 | + (−0.348 − 1.37i)2-s + (−1.71 − 0.213i)3-s + (−1.75 + 0.955i)4-s + (2.82 − 0.856i)5-s + (0.306 + 2.43i)6-s + (−0.0745 + 0.374i)7-s + (1.92 + 2.07i)8-s + (2.90 + 0.735i)9-s + (−2.15 − 3.57i)10-s + (−0.231 + 2.34i)11-s + (3.22 − 1.26i)12-s + (3.12 + 0.949i)13-s + (0.539 − 0.0285i)14-s + (−5.03 + 0.868i)15-s + (2.17 − 3.35i)16-s + (0.745 − 0.308i)17-s + ⋯ |
| L(s) = 1 | + (−0.246 − 0.969i)2-s + (−0.992 − 0.123i)3-s + (−0.878 + 0.477i)4-s + (1.26 − 0.383i)5-s + (0.124 + 0.992i)6-s + (−0.0281 + 0.141i)7-s + (0.679 + 0.733i)8-s + (0.969 + 0.245i)9-s + (−0.682 − 1.12i)10-s + (−0.0696 + 0.707i)11-s + (0.930 − 0.365i)12-s + (0.868 + 0.263i)13-s + (0.144 − 0.00761i)14-s + (−1.30 + 0.224i)15-s + (0.543 − 0.839i)16-s + (0.180 − 0.0748i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.844693 - 0.626468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.844693 - 0.626468i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.348 + 1.37i)T \) |
| 3 | \( 1 + (1.71 + 0.213i)T \) |
| good | 5 | \( 1 + (-2.82 + 0.856i)T + (4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (0.0745 - 0.374i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (0.231 - 2.34i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-3.12 - 0.949i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-0.745 + 0.308i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.936 + 0.500i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (-5.22 + 7.82i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.407 + 4.14i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (2.67 - 2.67i)T - 31iT^{2} \) |
| 37 | \( 1 + (5.34 + 2.85i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (-1.74 + 2.61i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-5.57 - 6.78i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-10.9 + 4.55i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.956 - 9.71i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (11.2 - 3.40i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 2.27i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-1.44 - 1.18i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (-0.884 + 4.44i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.73 + 0.742i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (0.353 - 0.853i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-6.40 + 3.42i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (5.43 - 3.63i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (8.45 + 8.45i)T + 97iT^{2} \) |
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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
−10.90394771171847691443366835176, −10.49949227558618404408833717847, −9.466674426739102887302880192777, −8.857910475934367935762468884682, −7.36676950532621547375013686808, −6.13702586664251838160603619882, −5.21871093538445285464624399478, −4.25608713209448005345309470738, −2.37259729291375626537457638453, −1.16086995552761891821186699148,
1.26649606591024713729828869996, 3.67727413393772236728905971721, 5.33893508645549524067030533169, 5.71006992309003470003812637439, 6.60877219719108295518640846488, 7.49835547176275740049279890208, 8.904690174270741676511338827583, 9.659712819927723439752057908612, 10.55597074906317172490335237872, 11.14004529853228998361835681425
