L(s) = 1 | + 5.48i·3-s + (53.0 + 17.7i)5-s + 188. i·7-s + 212.·9-s − 501.·11-s + 1.06e3i·13-s + (−97.5 + 290. i)15-s − 29.5i·17-s − 1.57e3·19-s − 1.03e3·21-s − 1.29e3i·23-s + (2.49e3 + 1.88e3i)25-s + 2.50e3i·27-s − 3.58e3·29-s + 3.52e3·31-s + ⋯ |
L(s) = 1 | + 0.352i·3-s + (0.948 + 0.317i)5-s + 1.45i·7-s + 0.875·9-s − 1.25·11-s + 1.74i·13-s + (−0.111 + 0.333i)15-s − 0.0248i·17-s − 1.00·19-s − 0.513·21-s − 0.510i·23-s + (0.798 + 0.602i)25-s + 0.660i·27-s − 0.791·29-s + 0.659·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.774069611\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.774069611\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-53.0 - 17.7i)T \) |
good | 3 | \( 1 - 5.48iT - 243T^{2} \) |
| 7 | \( 1 - 188. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 501.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.06e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 29.5iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.41e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.26e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.50e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.73e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.02e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.76e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.99e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.84e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.10e5iT - 8.58e9T^{2} \) |
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
−11.03408554906375990958123337230, −10.24847822952122974176088397485, −9.299727194685378596052457131480, −8.744794694829571472374079642015, −7.25708565278769887083361154480, −6.23349531795404343008837250115, −5.34590185999124840864971943844, −4.26864425932921979508012515636, −2.53068781389446046473730807228, −1.89598081998202538454240165060,
0.43930909774794211219689381578, 1.46593884147138193218684711683, 2.88556525985170918819418590568, 4.36590640246297794449076848368, 5.39265688890241174913065584630, 6.51772568102539708296274917688, 7.58900858195665072660384763966, 8.202261209366014119885732467954, 9.861584432457010848001108945203, 10.23407660004386489537430049418