L(s) = 1 | + (−5.16 + 14.7i)3-s − 48.6i·5-s + 167. i·7-s + (−189. − 151. i)9-s − 319.·11-s − 152.·13-s + (715. + 251. i)15-s − 1.32e3i·17-s + 1.61e3i·19-s + (−2.46e3 − 865. i)21-s − 1.22e3·23-s + 756.·25-s + (3.21e3 − 2.00e3i)27-s − 4.54e3i·29-s + 5.35e3i·31-s + ⋯ |
L(s) = 1 | + (−0.331 + 0.943i)3-s − 0.870i·5-s + 1.29i·7-s + (−0.780 − 0.624i)9-s − 0.795·11-s − 0.250·13-s + (0.821 + 0.288i)15-s − 1.11i·17-s + 1.02i·19-s + (−1.22 − 0.428i)21-s − 0.481·23-s + 0.242·25-s + (0.848 − 0.529i)27-s − 1.00i·29-s + 1.00i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.110433580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110433580\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.16 - 14.7i)T \) |
good | 5 | \( 1 + 48.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 167. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 319.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 152.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.32e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.61e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.54e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.35e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 3.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.04e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.32e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 5.88e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.39e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.22e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.70e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.59e5T + 8.58e9T^{2} \) |
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
−10.28828689993218228473101285959, −9.572916347515162279544290591338, −8.751057172066800896245332502045, −8.020383414547398963959833051574, −6.35894510259608759045098130461, −5.23524303526616055532669208323, −4.97385540813007559368158970436, −3.45985261551115074177407963325, −2.22251133573968828559836176482, −0.38939813035703453798242213740,
0.798468649525055564408477252502, 2.17553550886623490368346371977, 3.34533945512222807616005540960, 4.74135529171882215886945882638, 6.03700392731252375015223517988, 6.94543935832454017506607444995, 7.49900319466254529591441535465, 8.414498674370336393319249102453, 9.946552528872504005982590412573, 10.84583794093515221758697159758