| L(s) = 1 | + 4.29i·5-s + 5.06i·11-s − 3.37i·13-s + 2.76i·17-s + 4.70·19-s − 4.83i·23-s − 13.4·25-s − 2.46·29-s − 5.69·31-s − 2.33·37-s − 5.14i·41-s + 13.0i·43-s − 5.35·47-s + 4.22·53-s − 21.7·55-s + ⋯ |
| L(s) = 1 | + 1.92i·5-s + 1.52i·11-s − 0.937i·13-s + 0.670i·17-s + 1.07·19-s − 1.00i·23-s − 2.69·25-s − 0.456·29-s − 1.02·31-s − 0.383·37-s − 0.804i·41-s + 1.98i·43-s − 0.781·47-s + 0.580·53-s − 2.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7624685654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7624685654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.29iT - 5T^{2} \) |
| 11 | \( 1 - 5.06iT - 11T^{2} \) |
| 13 | \( 1 + 3.37iT - 13T^{2} \) |
| 17 | \( 1 - 2.76iT - 17T^{2} \) |
| 19 | \( 1 - 4.70T + 19T^{2} \) |
| 23 | \( 1 + 4.83iT - 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 + 2.33T + 37T^{2} \) |
| 41 | \( 1 + 5.14iT - 41T^{2} \) |
| 43 | \( 1 - 13.0iT - 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 - 3.87iT - 61T^{2} \) |
| 67 | \( 1 - 4.76iT - 67T^{2} \) |
| 71 | \( 1 - 12.3iT - 71T^{2} \) |
| 73 | \( 1 + 11.5iT - 73T^{2} \) |
| 79 | \( 1 + 13.9iT - 79T^{2} \) |
| 83 | \( 1 + 7.32T + 83T^{2} \) |
| 89 | \( 1 + 14.0iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−8.052676169113394937505181348061, −7.48217770810834097619799172357, −7.10640829746016872046657667681, −6.34901288450616913301460119567, −5.73807903464256025156805364060, −4.80028701182343437893479251775, −3.87109659238748465688559305371, −3.16242017882259825489799057814, −2.49836074672839114483913709126, −1.64595592179805648204813733988,
0.18805878077539160651344737733, 1.13434277924406988275178192561, 1.87496522483734251925045749437, 3.27253584553188334345902422322, 3.89898551221260448311674552325, 4.77946519330592672927957720823, 5.47715801217575704838406492371, 5.73394857689199771225436252621, 6.88777689112456729375236112690, 7.66936717056637275882414776655
