L(s) = 1 | + (1.05 − 1.69i)2-s + (−1.77 − 3.58i)4-s + 4.88i·5-s − 2.64i·7-s + (−7.96 − 0.763i)8-s + (8.29 + 5.14i)10-s + 21.4·11-s − 13.0i·13-s + (−4.49 − 2.79i)14-s + (−9.69 + 12.7i)16-s + 0.234·17-s + 4.55·19-s + (17.5 − 8.66i)20-s + (22.6 − 36.4i)22-s − 10.9i·23-s + ⋯ |
L(s) = 1 | + (0.527 − 0.849i)2-s + (−0.443 − 0.896i)4-s + 0.976i·5-s − 0.377i·7-s + (−0.995 − 0.0954i)8-s + (0.829 + 0.514i)10-s + 1.95·11-s − 1.00i·13-s + (−0.321 − 0.199i)14-s + (−0.606 + 0.795i)16-s + 0.0138·17-s + 0.239·19-s + (0.875 − 0.433i)20-s + (1.02 − 1.65i)22-s − 0.476i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.294641309\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.294641309\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.05 + 1.69i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 5 | \( 1 - 4.88iT - 25T^{2} \) |
| 11 | \( 1 - 21.4T + 121T^{2} \) |
| 13 | \( 1 + 13.0iT - 169T^{2} \) |
| 17 | \( 1 - 0.234T + 289T^{2} \) |
| 19 | \( 1 - 4.55T + 361T^{2} \) |
| 23 | \( 1 + 10.9iT - 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 + 34.1iT - 961T^{2} \) |
| 37 | \( 1 + 54.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.84T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 21.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 34.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 63.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 47.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 71.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 159.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 90.4T + 9.40e3T^{2} \) |
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
−10.69190242766487478875768741136, −9.781289735155926148444566435915, −9.061753630866275912285978324674, −7.66288671864423736830864769078, −6.53034182545695799856755375713, −5.86513758375473920110415710205, −4.34009733118629345282828664837, −3.58424172681808421868784598852, −2.45792089511942666955344603573, −0.898644178928316210260062273938,
1.42828770692086209755690614137, 3.44338031176250378056477827803, 4.43335883169105157923724680959, 5.24644911062643316826225319976, 6.42510687762627683253190964479, 7.02014824015067475958514649081, 8.404874107941376782814966153694, 8.990839900342585971158112113256, 9.545234042000053658357684118675, 11.33252327812599872673547403963