| L(s) = 1 | + (−0.910 + 1.89i)2-s + (23.8 + 1.78i)3-s + (37.1 + 46.5i)4-s + (47.7 − 154. i)5-s + (−25.1 + 43.4i)6-s + (360. − 208. i)7-s + (−252. + 57.7i)8-s + (−154. − 23.3i)9-s + (249. + 231. i)10-s + (423. − 531. i)11-s + (803. + 1.17e3i)12-s + (−556. + 516. i)13-s + (65.2 + 870. i)14-s + (1.41e3 − 3.61e3i)15-s + (−727. + 3.18e3i)16-s + (6.47e3 − 1.99e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.113 + 0.236i)2-s + (0.883 + 0.0662i)3-s + (0.580 + 0.728i)4-s + (0.382 − 1.23i)5-s + (−0.116 + 0.201i)6-s + (1.05 − 0.606i)7-s + (−0.493 + 0.112i)8-s + (−0.212 − 0.0319i)9-s + (0.249 + 0.231i)10-s + (0.318 − 0.399i)11-s + (0.464 + 0.681i)12-s + (−0.253 + 0.234i)13-s + (0.0237 + 0.317i)14-s + (0.420 − 1.07i)15-s + (−0.177 + 0.778i)16-s + (1.31 − 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0155i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 - 0.0155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.69471 + 0.0209957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.69471 + 0.0209957i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.21e4 - 7.63e4i)T \) |
| good | 2 | \( 1 + (0.910 - 1.89i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-23.8 - 1.78i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (-47.7 + 154. i)T + (-1.29e4 - 8.80e3i)T^{2} \) |
| 7 | \( 1 + (-360. + 208. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-423. + 531. i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (556. - 516. i)T + (3.60e5 - 4.81e6i)T^{2} \) |
| 17 | \( 1 + (-6.47e3 + 1.99e3i)T + (1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (-0.579 - 3.84i)T + (-4.49e7 + 1.38e7i)T^{2} \) |
| 23 | \( 1 + (-7.15e3 - 1.82e4i)T + (-1.08e8 + 1.00e8i)T^{2} \) |
| 29 | \( 1 + (-8.76e3 + 657. i)T + (5.88e8 - 8.86e7i)T^{2} \) |
| 31 | \( 1 + (2.92e4 - 1.99e4i)T + (3.24e8 - 8.26e8i)T^{2} \) |
| 37 | \( 1 + (5.52e4 + 3.18e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (8.70e4 + 4.19e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-6.88e4 - 8.62e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (1.15e5 + 1.07e5i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-1.61e4 + 7.07e4i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-4.58e4 + 6.73e4i)T + (-1.88e10 - 4.79e10i)T^{2} \) |
| 67 | \( 1 + (-2.52e5 + 3.80e4i)T + (8.64e10 - 2.66e10i)T^{2} \) |
| 71 | \( 1 + (-1.51e5 - 5.96e4i)T + (9.39e10 + 8.71e10i)T^{2} \) |
| 73 | \( 1 + (1.87e5 + 2.01e5i)T + (-1.13e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (4.15e5 + 7.19e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (6.25e4 - 8.34e5i)T + (-3.23e11 - 4.87e10i)T^{2} \) |
| 89 | \( 1 + (8.18e4 + 6.13e3i)T + (4.91e11 + 7.40e10i)T^{2} \) |
| 97 | \( 1 + (6.57e5 - 8.23e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−14.56022226407430069981608536875, −13.70903233225292702876602822978, −12.37961044555702248935579198320, −11.27727374501224797597206183238, −9.320089527024780523212546915200, −8.383673310471141355540304943888, −7.45588911988770111991618382666, −5.30100081018073525779001751565, −3.47464677445635701412662238779, −1.52993463392597493935139847610,
1.88800378598459704145438481885, 2.93122336362543680930777504887, 5.55349892682987316795705996186, 7.02619238595172597326220334427, 8.485166635491950420550580561225, 9.965423316570469943295162623288, 10.92732528855383327750654109561, 12.08943253641730283942303753394, 14.11052393207543436095933851151, 14.70869721744122283543332013275
