| L(s) = 1 | + (1.39 + 0.245i)2-s + (1.06 − 1.26i)3-s + (1.87 + 0.684i)4-s + (−4.36 + 1.58i)5-s + (1.78 − 1.49i)6-s + (−2.18 − 3.77i)7-s + (2.44 + 1.41i)8-s + (1.09 + 6.18i)9-s + (−6.46 + 1.13i)10-s + (−2.86 + 4.96i)11-s + (2.85 − 1.64i)12-s + (−11.9 − 14.2i)13-s + (−2.10 − 5.79i)14-s + (−2.61 + 7.19i)15-s + (3.06 + 2.57i)16-s + (1.14 − 6.49i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (0.353 − 0.421i)3-s + (0.469 + 0.171i)4-s + (−0.872 + 0.317i)5-s + (0.297 − 0.249i)6-s + (−0.311 − 0.539i)7-s + (0.306 + 0.176i)8-s + (0.121 + 0.687i)9-s + (−0.646 + 0.113i)10-s + (−0.260 + 0.451i)11-s + (0.238 − 0.137i)12-s + (−0.920 − 1.09i)13-s + (−0.150 − 0.414i)14-s + (−0.174 + 0.479i)15-s + (0.191 + 0.160i)16-s + (0.0673 − 0.381i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0376i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42916 - 0.0268815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42916 - 0.0268815i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.245i)T \) |
| 19 | \( 1 + (-18.9 - 0.274i)T \) |
| good | 3 | \( 1 + (-1.06 + 1.26i)T + (-1.56 - 8.86i)T^{2} \) |
| 5 | \( 1 + (4.36 - 1.58i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (2.18 + 3.77i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (2.86 - 4.96i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (11.9 + 14.2i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 6.49i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (-23.7 - 8.65i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-3.49 + 0.616i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-25.7 + 14.8i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 62.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (41.6 - 49.6i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (77.6 - 28.2i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (2.10 + 11.9i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (26.6 - 73.1i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-18.4 - 3.25i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (65.9 + 23.9i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-43.9 + 7.75i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-9.63 - 26.4i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-45.3 - 38.0i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (19.7 - 23.5i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (81.1 + 140. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (11.7 + 13.9i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-132. - 23.3i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−15.78625668186195245721055936016, −14.90172882604745192598168495736, −13.64156470091348948126950579529, −12.73951114617869945896980713281, −11.46217226557651279757162555510, −10.09461374354218747827857003171, −7.81333473420720688463761004118, −7.20074765855937627427897542453, −4.98312976618801595131740133966, −3.08439462431192012993607026923,
3.29824773426863691597142525551, 4.81821138862777999223130520541, 6.76710328364940461034908522613, 8.547567433431406393000229671257, 9.886876507456652025525457127896, 11.67824589591449784541821790544, 12.32038464966886522481401921852, 13.84229079280823418413640741150, 15.06690299736719115771751329977, 15.72612124144041455787451660503
