L(s) = 1 | + (1.09 + 1.34i)3-s + (1.95 + 3.39i)5-s + (−0.606 + 2.93i)9-s + (3.19 + 1.84i)11-s + (−0.480 + 0.277i)13-s + (−2.41 + 6.33i)15-s + 5.83·17-s − 5.33i·19-s + (1.96 − 1.13i)23-s + (−5.16 + 8.94i)25-s + (−4.60 + 2.40i)27-s + (3.53 + 2.04i)29-s + (7.00 − 4.04i)31-s + (1.01 + 6.31i)33-s − 7.79·37-s + ⋯ |
L(s) = 1 | + (0.631 + 0.775i)3-s + (0.875 + 1.51i)5-s + (−0.202 + 0.979i)9-s + (0.964 + 0.556i)11-s + (−0.133 + 0.0769i)13-s + (−0.622 + 1.63i)15-s + 1.41·17-s − 1.22i·19-s + (0.410 − 0.237i)23-s + (−1.03 + 1.78i)25-s + (−0.886 + 0.461i)27-s + (0.656 + 0.379i)29-s + (1.25 − 0.726i)31-s + (0.177 + 1.09i)33-s − 1.28·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.775800350\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.775800350\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.09 - 1.34i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.95 - 3.39i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.19 - 1.84i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.480 - 0.277i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 + 5.33iT - 19T^{2} \) |
| 23 | \( 1 + (-1.96 + 1.13i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.53 - 2.04i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.00 + 4.04i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 + (3.59 + 6.22i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.754 - 1.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.0479iT - 53T^{2} \) |
| 59 | \( 1 + (4.45 + 7.71i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.03 + 3.48i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.587 + 1.01i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.71iT - 71T^{2} \) |
| 73 | \( 1 + 4.07iT - 73T^{2} \) |
| 79 | \( 1 + (-1.97 + 3.41i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.84 - 6.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 5.42T + 89T^{2} \) |
| 97 | \( 1 + (13.9 + 8.07i)T + (48.5 + 84.0i)T^{2} \) |
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
−9.620421776951967867273459556688, −9.032003720573000718957955943669, −7.950988472004299609998777636730, −7.04045528207082297995300862337, −6.50551823916029060526662203145, −5.45146046838084990338665347284, −4.53346613381046903985359112981, −3.37779944571588165910585396139, −2.83203792492312316236954275474, −1.78796362771335283523216435419,
1.11857787356516582090747262251, 1.50657208950268805323975477257, 2.95595253107974319018282504805, 3.95373015927698233990607465800, 5.11544610443997544802094068133, 5.88195039818092733972482664874, 6.54895589681237225910635085860, 7.69576877600684883372776678230, 8.478122907860758858004090752906, 8.803380769397776252265153658575