L(s) = 1 | − 13.2i·2-s − 15.5·3-s − 111.·4-s − 224.·5-s + 206. i·6-s + 460.·7-s + 629. i·8-s + 243·9-s + 2.97e3i·10-s − 1.56e3i·11-s + 1.73e3·12-s + 3.96e3i·13-s − 6.09e3i·14-s + 3.49e3·15-s + 1.20e3·16-s + 6.70e3·17-s + ⋯ |
L(s) = 1 | − 1.65i·2-s − 0.577·3-s − 1.74·4-s − 1.79·5-s + 0.956i·6-s + 1.34·7-s + 1.23i·8-s + 0.333·9-s + 2.97i·10-s − 1.17i·11-s + 1.00·12-s + 1.80i·13-s − 2.22i·14-s + 1.03·15-s + 0.294·16-s + 1.36·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.086948067\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.086948067\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 15.5T \) |
| 59 | \( 1 + (-1.93e5 + 6.85e4i)T \) |
good | 2 | \( 1 + 13.2iT - 64T^{2} \) |
| 5 | \( 1 + 224.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 460.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.56e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.96e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.70e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 8.70e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 7.02e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.83e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.52e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.95e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 7.06e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 9.87e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 8.70e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.97e3T + 2.21e10T^{2} \) |
| 61 | \( 1 - 7.58e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.05e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.51e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.46e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 1.88e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.71e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 6.09e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 9.98e5iT - 8.32e11T^{2} \) |
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
−11.24334792549988154734774452881, −10.88791432443482948750412160872, −9.297879415892085333445923324077, −8.273561190302946794826940666288, −7.26009623223043034703006244752, −5.17213990600473767189423848316, −4.16248800936267264711825327868, −3.37523943220920543574462257543, −1.52601733638022973632133698139, −0.54587535652993949949902223681,
0.819993813689550692540746581290, 3.74104428764297596122968453673, 4.96086826216396749113794388009, 5.48385571869442264319322787196, 7.35367504359826419211093883908, 7.64250947944388879471089762490, 8.227422485829794006896105835562, 9.885224640112220739501483173128, 11.29107312943903880432147476184, 11.95386807580743513198444154880