L(s) = 1 | + (0.00869 + 0.0150i)5-s + (−2.50 + 0.851i)7-s + (4.13 + 2.38i)11-s + (−1.31 + 0.759i)13-s + 1.91·17-s + 6.45i·19-s + (3.82 − 2.20i)23-s + (2.49 − 4.32i)25-s + (−1.73 − 1.00i)29-s + (−6.51 + 3.76i)31-s + (−0.0346 − 0.0303i)35-s − 6.10·37-s + (−4.67 − 8.10i)41-s + (1.46 − 2.53i)43-s + (−1.45 + 2.52i)47-s + ⋯ |
L(s) = 1 | + (0.00389 + 0.00673i)5-s + (−0.946 + 0.321i)7-s + (1.24 + 0.719i)11-s + (−0.364 + 0.210i)13-s + 0.463·17-s + 1.48i·19-s + (0.797 − 0.460i)23-s + (0.499 − 0.865i)25-s + (−0.322 − 0.186i)29-s + (−1.17 + 0.676i)31-s + (−0.00585 − 0.00512i)35-s − 1.00·37-s + (−0.730 − 1.26i)41-s + (0.223 − 0.387i)43-s + (−0.212 + 0.368i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147151208\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147151208\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.50 - 0.851i)T \) |
good | 5 | \( 1 + (-0.00869 - 0.0150i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.13 - 2.38i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.31 - 0.759i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 - 6.45iT - 19T^{2} \) |
| 23 | \( 1 + (-3.82 + 2.20i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.73 + 1.00i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.51 - 3.76i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 + (4.67 + 8.10i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.46 + 2.53i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.45 - 2.52i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.94iT - 53T^{2} \) |
| 59 | \( 1 + (4.08 + 7.07i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.484 + 0.279i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.69 - 6.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.6iT - 71T^{2} \) |
| 73 | \( 1 - 12.8iT - 73T^{2} \) |
| 79 | \( 1 + (4.75 - 8.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.67 - 11.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.00T + 89T^{2} \) |
| 97 | \( 1 + (-4.09 - 2.36i)T + (48.5 + 84.0i)T^{2} \) |
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
−8.974514190477976732145470714641, −8.398497127516302228486726018832, −7.15148150888742946478960193204, −6.87884560941950141149403569454, −5.95800088698157963760362147514, −5.22279051635295805169370875524, −4.08643381814916516944835029904, −3.52011823024460964045026125635, −2.39802907483878758272582852531, −1.32578487597393293780653033584,
0.37520643942762524111942017993, 1.57858148977571582254804926927, 3.13843047014925433188738530805, 3.41747950479761705072618987223, 4.58577772415141814709239604481, 5.44930244816455605429553289094, 6.32177202244228526922458175183, 6.97353874567323894431128548693, 7.52529356929044378030838026987, 8.736121006595522691819124052064