L(s) = 1 | + (2.81 + 1.04i)3-s + (−4.33 − 2.48i)5-s + 10.8i·7-s + (6.83 + 5.85i)9-s − 15.2·11-s + 1.83·13-s + (−9.62 − 11.5i)15-s + 20.4·17-s − 13.6i·19-s + (−11.3 + 30.6i)21-s − 19.4·23-s + (12.6 + 21.5i)25-s + (13.1 + 23.6i)27-s − 24.5·29-s − 50.8·31-s + ⋯ |
L(s) = 1 | + (0.937 + 0.347i)3-s + (−0.867 − 0.496i)5-s + 1.55i·7-s + (0.758 + 0.651i)9-s − 1.39·11-s + 0.141·13-s + (−0.641 − 0.767i)15-s + 1.20·17-s − 0.719i·19-s + (−0.539 + 1.45i)21-s − 0.845·23-s + (0.506 + 0.862i)25-s + (0.485 + 0.874i)27-s − 0.847·29-s − 1.64·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0888i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7386541417\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7386541417\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.81 - 1.04i)T \) |
| 5 | \( 1 + (4.33 + 2.48i)T \) |
good | 7 | \( 1 - 10.8iT - 49T^{2} \) |
| 11 | \( 1 + 15.2T + 121T^{2} \) |
| 13 | \( 1 - 1.83T + 169T^{2} \) |
| 17 | \( 1 - 20.4T + 289T^{2} \) |
| 19 | \( 1 + 13.6iT - 361T^{2} \) |
| 23 | \( 1 + 19.4T + 529T^{2} \) |
| 29 | \( 1 + 24.5T + 841T^{2} \) |
| 31 | \( 1 + 50.8T + 961T^{2} \) |
| 37 | \( 1 + 16.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 26.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 63.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 31.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 81.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 50.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 135. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 81.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 75.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 124. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 58.1iT - 9.40e3T^{2} \) |
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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
−10.04416758387103866993247350260, −9.223618904767197596140544337396, −8.528420810657767084172348532543, −7.975848379151405545606919817116, −7.21640028670239121748144249287, −5.51442648835383969901890781943, −5.13992281483132623763766390010, −3.80973912927470442511408086085, −2.94714121078196378125185741025, −1.93340580196035278644089935745,
0.19542558009357851189743621934, 1.69923515186900184043803248642, 3.26890470427646574054143673364, 3.63879581900456624070729962057, 4.75955766656859668998518681747, 6.20456853538149349932551184853, 7.38262585426848939264896078774, 7.66898297944749564533222947074, 8.167183094021447982889393940617, 9.524757479852535323944677689164