| 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
−15.72612124144041455787451660503, −15.06690299736719115771751329977, −13.84229079280823418413640741150, −12.32038464966886522481401921852, −11.67824589591449784541821790544, −9.886876507456652025525457127896, −8.547567433431406393000229671257, −6.76710328364940461034908522613, −4.81821138862777999223130520541, −3.29824773426863691597142525551,
3.08439462431192012993607026923, 4.98312976618801595131740133966, 7.20074765855937627427897542453, 7.81333473420720688463761004118, 10.09461374354218747827857003171, 11.46217226557651279757162555510, 12.73951114617869945896980713281, 13.64156470091348948126950579529, 14.90172882604745192598168495736, 15.78625668186195245721055936016
