| L(s) = 1 | + (−7.49 − 32.8i)2-s + (195. + 181. i)3-s + (−559. + 269. i)4-s + (2.54e3 − 383. i)5-s + (4.48e3 − 7.77e3i)6-s + (−1.78e3 − 3.09e3i)7-s + (2.29e3 + 2.88e3i)8-s + (3.84e3 + 5.12e4i)9-s + (−3.16e4 − 8.06e4i)10-s + (−5.52e3 − 2.65e3i)11-s + (−1.58e5 − 4.88e4i)12-s + (5.02e4 − 1.28e5i)13-s + (−8.83e4 + 8.19e4i)14-s + (5.66e5 + 3.86e5i)15-s + (−1.20e5 + 1.51e5i)16-s + (5.20e5 + 7.84e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.331 − 1.45i)2-s + (1.39 + 1.29i)3-s + (−1.09 + 0.526i)4-s + (1.82 − 0.274i)5-s + (1.41 − 2.44i)6-s + (−0.281 − 0.487i)7-s + (0.198 + 0.248i)8-s + (0.195 + 2.60i)9-s + (−1.00 − 2.55i)10-s + (−0.113 − 0.0547i)11-s + (−2.20 − 0.680i)12-s + (0.488 − 1.24i)13-s + (−0.614 + 0.570i)14-s + (2.89 + 1.97i)15-s + (−0.461 + 0.578i)16-s + (1.51 + 0.227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.98746 - 1.62623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.98746 - 1.62623i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.72e7 + 1.43e7i)T \) |
| good | 2 | \( 1 + (7.49 + 32.8i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-195. - 181. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-2.54e3 + 383. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (1.78e3 + 3.09e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (5.52e3 + 2.65e3i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (-5.02e4 + 1.28e5i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (-5.20e5 - 7.84e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-745. + 9.95e3i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (6.71e5 - 4.57e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (1.32e6 - 1.22e6i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (-1.87e5 - 5.79e4i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (2.34e6 - 4.06e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (1.11e5 + 4.86e5i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (2.77e7 - 1.33e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (3.40e7 + 8.68e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (-4.63e6 + 5.80e6i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (8.25e7 - 2.54e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (9.27e6 - 1.23e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (-1.20e8 - 8.24e7i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (1.22e8 - 3.12e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (3.50e7 + 6.06e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.44e8 + 1.33e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-2.64e8 - 2.45e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (6.20e8 + 2.98e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
| 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
−13.60410841098881459372422861959, −12.89625154128443241647253954532, −10.61014392493528180836828294833, −10.06663707389197829401614339474, −9.492236799885489230723602282354, −8.335535644992798435968939351615, −5.42028247178716233909496121531, −3.62486954024078085107587273991, −2.69371515813331435728559906843, −1.42111319422386635321377458758,
1.53537473249116282534632879174, 2.71134883698807041420447560219, 5.89967005320784323215320329986, 6.57199055314005697354240190214, 7.73693728055422751137025950491, 9.023078123616035874090121587340, 9.561125664777237710574911410060, 12.39691734031341414165464730768, 13.74065448517302168936187546948, 14.07329400956624634996029350770
