| L(s) = 1 | + (6.34 − 13.1i)2-s + (10.7 − 15.7i)3-s + (−93.3 − 117. i)4-s + (−134. − 144. i)5-s + (−139. − 241. i)6-s + (570. + 329. i)7-s + (−1.22e3 + 278. i)8-s + (133. + 339. i)9-s + (−2.75e3 + 851. i)10-s + (745. − 934. i)11-s + (−2.85e3 + 213. i)12-s + (815. + 251. i)13-s + (7.95e3 − 5.42e3i)14-s + (−3.73e3 + 562. i)15-s + (−1.94e3 + 8.51e3i)16-s + (−4.60e3 − 4.27e3i)17-s + ⋯ |
| L(s) = 1 | + (0.792 − 1.64i)2-s + (0.398 − 0.584i)3-s + (−1.45 − 1.82i)4-s + (−1.07 − 1.15i)5-s + (−0.646 − 1.11i)6-s + (1.66 + 0.959i)7-s + (−2.38 + 0.544i)8-s + (0.182 + 0.465i)9-s + (−2.75 + 0.851i)10-s + (0.559 − 0.701i)11-s + (−1.64 + 0.123i)12-s + (0.371 + 0.114i)13-s + (2.89 − 1.97i)14-s + (−1.10 + 0.166i)15-s + (−0.474 + 2.07i)16-s + (−0.936 − 0.869i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.615554 + 2.43449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.615554 + 2.43449i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.30e4 - 4.84e4i)T \) |
| good | 2 | \( 1 + (-6.34 + 13.1i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-10.7 + 15.7i)T + (-266. - 678. i)T^{2} \) |
| 5 | \( 1 + (134. + 144. i)T + (-1.16e3 + 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-570. - 329. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-745. + 934. i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-815. - 251. i)T + (3.98e6 + 2.71e6i)T^{2} \) |
| 17 | \( 1 + (4.60e3 + 4.27e3i)T + (1.80e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (1.27e3 + 500. i)T + (3.44e7 + 3.19e7i)T^{2} \) |
| 23 | \( 1 + (9.95e3 + 1.49e3i)T + (1.41e8 + 4.36e7i)T^{2} \) |
| 29 | \( 1 + (-1.56e4 - 2.29e4i)T + (-2.17e8 + 5.53e8i)T^{2} \) |
| 31 | \( 1 + (2.03e3 + 2.71e4i)T + (-8.77e8 + 1.32e8i)T^{2} \) |
| 37 | \( 1 + (-1.54e4 + 8.90e3i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-8.65e3 - 4.16e3i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (6.28e4 + 7.87e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-1.45e5 + 4.47e4i)T + (1.83e10 - 1.24e10i)T^{2} \) |
| 59 | \( 1 + (-5.01e4 + 2.19e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.77e5 - 1.32e4i)T + (5.09e10 + 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-2.03e4 + 5.18e4i)T + (-6.63e10 - 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-1.21e4 - 8.03e4i)T + (-1.22e11 + 3.77e10i)T^{2} \) |
| 73 | \( 1 + (1.55e5 - 5.04e5i)T + (-1.25e11 - 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-1.10e5 + 1.91e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-5.53e5 - 3.77e5i)T + (1.19e11 + 3.04e11i)T^{2} \) |
| 89 | \( 1 + (5.03e4 - 7.38e4i)T + (-1.81e11 - 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-6.57e5 + 8.24e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
| 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
−13.67301924338139521750438991410, −12.59241464940279664395089137976, −11.67296098423022429863767257949, −11.16596571584019492221880631510, −8.933784560168441425915365940073, −8.189539062318841700445827540290, −5.15887324372249410868524942475, −4.24077877173428304534455702698, −2.20971294232169612647724654730, −1.00265574462836839337192184519,
3.93228604367283568473413644739, 4.36124598596309612484416999713, 6.59109975814480595063940597592, 7.57053897504236956601207056911, 8.481981229214622207491925763068, 10.61826147441761799689275900618, 11.97772371415870764864415278857, 13.80173598551537548244594341492, 14.74095434310167850341469556764, 14.99708997260013379108047517653
