| L(s) = 1 | + (3.31 + 0.887i)2-s + (5.15 + 0.641i)3-s + (3.24 + 1.87i)4-s + (−9.09 + 9.09i)5-s + (16.5 + 6.69i)6-s + (−7.29 − 27.2i)7-s + (−10.3 − 10.3i)8-s + (26.1 + 6.61i)9-s + (−38.2 + 22.0i)10-s + (−8.81 + 32.8i)11-s + (15.5 + 11.7i)12-s + (46.8 + 0.722i)13-s − 96.6i·14-s + (−52.7 + 41.0i)15-s + (−39.9 − 69.2i)16-s + (6.02 − 10.4i)17-s + ⋯ |
| L(s) = 1 | + (1.17 + 0.313i)2-s + (0.992 + 0.123i)3-s + (0.405 + 0.234i)4-s + (−0.813 + 0.813i)5-s + (1.12 + 0.455i)6-s + (−0.393 − 1.47i)7-s + (−0.455 − 0.455i)8-s + (0.969 + 0.245i)9-s + (−1.20 + 0.697i)10-s + (−0.241 + 0.901i)11-s + (0.373 + 0.282i)12-s + (0.999 + 0.0154i)13-s − 1.84i·14-s + (−0.908 + 0.707i)15-s + (−0.624 − 1.08i)16-s + (0.0860 − 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.32662 + 0.499591i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32662 + 0.499591i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.15 - 0.641i)T \) |
| 13 | \( 1 + (-46.8 - 0.722i)T \) |
| good | 2 | \( 1 + (-3.31 - 0.887i)T + (6.92 + 4i)T^{2} \) |
| 5 | \( 1 + (9.09 - 9.09i)T - 125iT^{2} \) |
| 7 | \( 1 + (7.29 + 27.2i)T + (-297. + 171.5i)T^{2} \) |
| 11 | \( 1 + (8.81 - 32.8i)T + (-1.15e3 - 665.5i)T^{2} \) |
| 17 | \( 1 + (-6.02 + 10.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (82.7 - 22.1i)T + (5.94e3 - 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-51.1 - 88.6i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (43.1 - 24.9i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (108. + 108. i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (-369. - 98.9i)T + (4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-94.7 - 25.3i)T + (5.96e4 + 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-66.6 - 38.4i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (222. + 222. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + 276. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (52.4 - 14.0i)T + (1.77e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-156. + 270. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-26.8 + 100. i)T + (-2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 + (288. + 1.07e3i)T + (-3.09e5 + 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-255. + 255. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 603.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (616. - 616. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (291. - 1.08e3i)T + (-6.10e5 - 3.52e5i)T^{2} \) |
| 97 | \( 1 + (1.03e3 - 278. i)T + (7.90e5 - 4.56e5i)T^{2} \) |
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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
−15.26830051641955502359290172527, −14.69873967182031232183922704437, −13.55922217962040956253796040734, −12.89124183757672363716026619003, −11.00358913005154314584695974569, −9.697413062058201080744476257689, −7.69196600452101911030003414027, −6.74622511240657294050598697078, −4.25967983742440832691659752316, −3.45197283858954206933457175684,
2.82687267558593252383181860981, 4.23452071196875234611869763134, 5.93914387034416213526130623467, 8.445177109055838334995707963706, 8.860535127168303603849689317276, 11.31390137886768031044646932961, 12.65093035135278993456436408600, 12.97327155499120147395099451088, 14.41672232846675670463335801911, 15.39995871523422966410402947511
