| L(s) = 1 | + (8 + 13.8i)2-s + (37.7 + 65.4i)3-s + (−127. + 221. i)4-s + (−802. − 1.39e3i)5-s + (−604. + 1.04e3i)6-s + 1.58e3·7-s − 4.09e3·8-s + (6.98e3 − 1.21e4i)9-s + (1.28e4 − 2.22e4i)10-s + 2.29e4·11-s − 1.93e4·12-s + (7.40e4 − 1.28e5i)13-s + (1.26e4 + 2.19e4i)14-s + (6.06e4 − 1.05e5i)15-s + (−3.27e4 − 5.67e4i)16-s + (2.27e5 + 3.93e5i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.269 + 0.466i)3-s + (−0.249 + 0.433i)4-s + (−0.574 − 0.995i)5-s + (−0.190 + 0.329i)6-s + 0.249·7-s − 0.353·8-s + (0.355 − 0.615i)9-s + (0.406 − 0.703i)10-s + 0.471·11-s − 0.269·12-s + (0.719 − 1.24i)13-s + (0.0882 + 0.152i)14-s + (0.309 − 0.535i)15-s + (−0.125 − 0.216i)16-s + (0.659 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0668i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.28924 - 0.0766431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28924 - 0.0766431i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 - 13.8i)T \) |
| 19 | \( 1 + (5.62e5 - 8.25e4i)T \) |
| good | 3 | \( 1 + (-37.7 - 65.4i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (802. + 1.39e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 - 1.58e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.29e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-7.40e4 + 1.28e5i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-2.27e5 - 3.93e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (-1.04e6 + 1.80e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.07e6 - 1.86e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 - 3.63e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.01e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (7.28e6 + 1.26e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.16e7 + 2.01e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-1.58e7 + 2.74e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (2.60e7 - 4.51e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (8.43e7 + 1.46e8i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.37e7 - 5.84e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (5.27e7 - 9.14e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (5.27e7 + 9.13e7i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-4.53e7 - 7.85e7i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (5.02e7 + 8.70e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 5.84e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.88e8 + 4.99e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-4.47e8 - 7.74e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−14.69506134943003583183241215920, −12.96103280152915206677580855974, −12.30452847468184005072453956897, −10.51416959851942206203741045367, −8.845414402199875488396169025252, −8.109962561353364301110714064489, −6.25635613899340578638270540043, −4.66087091529358545117434841733, −3.60178337047165907387846722625, −0.829612006890533559189950114411,
1.46115901389038491751261584149, 2.96840505022850825840962237212, 4.46139775769700275263718399258, 6.55577580489051849216816057328, 7.76227375458132387545256065323, 9.463115796514080992692326646273, 11.05073628465816867390972774212, 11.65390573300999009588968813287, 13.24186584093703983104373336368, 14.12557797922968648063775179538
