L(s) = 1 | + (−6.25 + 6.25i)2-s + (−11.2 + 11.2i)3-s − 14.2i·4-s + 192. i·5-s − 140. i·6-s − 54.6·7-s + (−311. − 311. i)8-s + 477. i·9-s + (−1.20e3 − 1.20e3i)10-s + (785. − 785. i)11-s + (159. + 159. i)12-s − 3.66e3i·13-s + (341. − 341. i)14-s + (−2.16e3 − 2.16e3i)15-s + 4.80e3·16-s + (−4.65e3 + 4.65e3i)17-s + ⋯ |
L(s) = 1 | + (−0.781 + 0.781i)2-s + (−0.415 + 0.415i)3-s − 0.222i·4-s + 1.54i·5-s − 0.649i·6-s − 0.159·7-s + (−0.607 − 0.607i)8-s + 0.655i·9-s + (−1.20 − 1.20i)10-s + (0.590 − 0.590i)11-s + (0.0923 + 0.0923i)12-s − 1.66i·13-s + (0.124 − 0.124i)14-s + (−0.640 − 0.640i)15-s + 1.17·16-s + (−0.947 + 0.947i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.236019 - 0.420518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.236019 - 0.420518i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (8.53e3 - 2.28e4i)T \) |
good | 2 | \( 1 + (6.25 - 6.25i)T - 64iT^{2} \) |
| 3 | \( 1 + (11.2 - 11.2i)T - 729iT^{2} \) |
| 5 | \( 1 - 192. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 54.6T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-785. + 785. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + 3.66e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (4.65e3 - 4.65e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + (4.78e3 - 4.78e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 - 2.03e4T + 1.48e8T^{2} \) |
| 31 | \( 1 + (5.99e3 - 5.99e3i)T - 8.87e8iT^{2} \) |
| 37 | \( 1 + (1.39e4 + 1.39e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + (3.25e4 + 3.25e4i)T + 4.75e9iT^{2} \) |
| 43 | \( 1 + (3.92e4 - 3.92e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (-7.76e3 - 7.76e3i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 - 1.97e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.92e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + (9.47e4 - 9.47e4i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 - 1.47e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 5.18e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (1.79e5 + 1.79e5i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 + (3.23e5 - 3.23e5i)T - 2.43e11iT^{2} \) |
| 83 | \( 1 - 2.78e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-3.97e5 + 3.97e5i)T - 4.96e11iT^{2} \) |
| 97 | \( 1 + (2.54e5 + 2.54e5i)T + 8.32e11iT^{2} \) |
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
−16.75985190996216091730142121833, −15.47397710749433615361583614898, −14.75911180010395275746229895297, −12.97321343092038386409535370897, −10.98763509134804589350562442705, −10.32052205812473088617612958047, −8.520950555046526628950929400593, −7.15469730148552357237854043568, −5.96565611089486158442711433592, −3.28995355296540714478465797924,
0.36429902815028652294938213427, 1.68189550912806283903797966457, 4.70362781175108423110305018721, 6.66462791840985011573401707996, 9.070653355368619917541639132922, 9.248094828006066964386743220246, 11.39913291703435746492975358997, 12.08930304250154002831100696617, 13.31158313420934487603413085278, 15.13265568174024776650719238757