| L(s) = 1 | + (32 + 55.4i)2-s + (−746. + 1.29e3i)3-s + (−2.04e3 + 3.54e3i)4-s + (−7.17e3 − 1.24e4i)5-s − 9.55e4·6-s + (−3.02e5 + 7.33e4i)7-s − 2.62e5·8-s + (−3.18e5 − 5.51e5i)9-s + (4.59e5 − 7.95e5i)10-s + (2.14e6 − 3.71e6i)11-s + (−3.05e6 − 5.29e6i)12-s + 5.50e6·13-s + (−1.37e7 − 1.44e7i)14-s + 2.14e7·15-s + (−8.38e6 − 1.45e7i)16-s + (1.98e7 − 3.43e7i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.591 + 1.02i)3-s + (−0.249 + 0.433i)4-s + (−0.205 − 0.355i)5-s − 0.836·6-s + (−0.971 + 0.235i)7-s − 0.353·8-s + (−0.199 − 0.345i)9-s + (0.145 − 0.251i)10-s + (0.364 − 0.631i)11-s + (−0.295 − 0.512i)12-s + 0.316·13-s + (−0.487 − 0.511i)14-s + 0.486·15-s + (−0.125 − 0.216i)16-s + (0.199 − 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.0579731 - 0.0486466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0579731 - 0.0486466i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-32 - 55.4i)T \) |
| 7 | \( 1 + (3.02e5 - 7.33e4i)T \) |
| good | 3 | \( 1 + (746. - 1.29e3i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (7.17e3 + 1.24e4i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-2.14e6 + 3.71e6i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 - 5.50e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-1.98e7 + 3.43e7i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (2.25e7 + 3.89e7i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (3.77e8 + 6.53e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + 2.46e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (3.06e9 - 5.31e9i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (-9.52e9 - 1.64e10i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + 4.23e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 7.42e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (4.03e10 + 6.98e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-1.10e11 + 1.91e11i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (2.29e11 - 3.98e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (3.23e10 + 5.61e10i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-2.07e11 + 3.58e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + 6.38e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (2.92e11 - 5.07e11i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (1.22e12 + 2.12e12i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 - 3.32e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-1.39e12 - 2.41e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 + 7.80e12T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.35176510862073773149419981943, −15.05569516588611963238119180524, −13.35421306079806513940599400455, −11.84514646932390625610178713373, −10.17230203723905575563776221549, −8.683851147460085844168503535326, −6.45516285448462248752282436954, −5.03146044966849350581870089078, −3.55525541228603105073194405151, −0.02927334630808015149752848512,
1.61242001388358858830932104211, 3.63402336398901053449189313561, 5.98511076326541554076456885122, 7.24915833341416898654297228712, 9.681958098997286179463434815863, 11.31127581068134429806853173817, 12.50001779292121956278034011576, 13.39164320016711891016990402553, 15.10157753912557984851178846007, 16.90725083771873325312095333041
