L(s) = 1 | + 4.24·5-s − 4·7-s + 16.9·11-s − 8i·13-s − 12.7i·17-s − 16i·19-s − 16.9i·23-s − 7.00·25-s + 4.24·29-s − 44·31-s − 16.9·35-s − 34i·37-s − 46.6i·41-s + 40i·43-s + 84.8i·47-s + ⋯ |
L(s) = 1 | + 0.848·5-s − 0.571·7-s + 1.54·11-s − 0.615i·13-s − 0.748i·17-s − 0.842i·19-s − 0.737i·23-s − 0.280·25-s + 0.146·29-s − 1.41·31-s − 0.484·35-s − 0.918i·37-s − 1.13i·41-s + 0.930i·43-s + 1.80i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.885773801\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.885773801\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4.24T + 25T^{2} \) |
| 7 | \( 1 + 4T + 49T^{2} \) |
| 11 | \( 1 - 16.9T + 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 + 16iT - 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 4.24T + 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 + 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 50iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16T + 5.32e3T^{2} \) |
| 79 | \( 1 - 76T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.960746800387751010161628477801, −7.77472761363281407135330571091, −6.89864841133309236935315499281, −6.30849906416908693412437830873, −5.59257524818157604085686642532, −4.62112821039282335495805284946, −3.63980142940457781455487591228, −2.71470673711931664214546839366, −1.65404570545062085024827180489, −0.44296085759583310367773924084,
1.36432036659894944600216157795, 2.01152365581565939186641765976, 3.48137153723380943899903318778, 3.98136935092309155945015115159, 5.22190377123343109696560154071, 6.10478465648521390521766871097, 6.51429747001197841894099794753, 7.35994905536826164288172788208, 8.468992135182552435052986414140, 9.127628475774203908764263851747