L(s) = 1 | + (2.23 − 2i)3-s + (−2.23 − 4.47i)5-s + 8i·7-s + (1.00 − 8.94i)9-s + 8.94i·11-s − 12i·13-s + (−13.9 − 5.52i)15-s − 31.3·17-s − 6·19-s + (16 + 17.8i)21-s − 4.47·23-s + (−15.0 + 20.0i)25-s + (−15.6 − 22.0i)27-s − 26.8i·29-s − 34·31-s + ⋯ |
L(s) = 1 | + (0.745 − 0.666i)3-s + (−0.447 − 0.894i)5-s + 1.14i·7-s + (0.111 − 0.993i)9-s + 0.813i·11-s − 0.923i·13-s + (−0.929 − 0.368i)15-s − 1.84·17-s − 0.315·19-s + (0.761 + 0.851i)21-s − 0.194·23-s + (−0.600 + 0.800i)25-s + (−0.579 − 0.814i)27-s − 0.925i·29-s − 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4226066754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4226066754\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.23 + 2i)T \) |
| 5 | \( 1 + (2.23 + 4.47i)T \) |
good | 7 | \( 1 - 8iT - 49T^{2} \) |
| 11 | \( 1 - 8.94iT - 121T^{2} \) |
| 13 | \( 1 + 12iT - 169T^{2} \) |
| 17 | \( 1 + 31.3T + 289T^{2} \) |
| 19 | \( 1 + 6T + 361T^{2} \) |
| 23 | \( 1 + 4.47T + 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 34T + 961T^{2} \) |
| 37 | \( 1 - 44iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 17.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 4.47T + 2.20e3T^{2} \) |
| 53 | \( 1 + 40.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 98.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74T + 3.72e3T^{2} \) |
| 67 | \( 1 - 92iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 56iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 78T + 6.24e3T^{2} \) |
| 83 | \( 1 + 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 32iT - 9.40e3T^{2} \) |
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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
−9.081636857067884962631281360326, −8.562143969444075783951828793479, −7.88814255911263633224280358352, −6.93492592047820895183909839816, −5.95104285922947198801673972190, −4.91065164978837448556353049093, −3.92233544001999396905447563872, −2.59798153272257526442868259051, −1.76549268082112390424949835283, −0.11105871862258896519504784525,
1.99444385430759652777462243598, 3.19218917468931507695500974511, 4.00398209765649515506036594233, 4.62348672661812909043474462784, 6.21869073284153128396092482381, 7.07252229252093714449022158686, 7.72561792248015513344065362736, 8.774617972435668936782916587053, 9.343977000041187132292454717877, 10.58058231844079893267857716335