| L(s) = 1 | − 1.41i·5-s − 3.16·7-s + 4.47i·11-s + 4.47i·13-s − 6.32·17-s − 2.82i·19-s + 4·23-s + 2.99·25-s − 4.24i·29-s − 3.16·31-s + 4.47i·35-s − 4.47i·37-s + 6.32·41-s + 8.48i·43-s − 12·47-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s − 1.19·7-s + 1.34i·11-s + 1.24i·13-s − 1.53·17-s − 0.648i·19-s + 0.834·23-s + 0.599·25-s − 0.787i·29-s − 0.567·31-s + 0.755i·35-s − 0.735i·37-s + 0.987·41-s + 1.29i·43-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8310273600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8310273600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 - 4.47iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.240370350843301696822302497379, −7.13371999024041449366411315549, −6.77364882496331221957506989820, −6.17428228609004729676627786957, −4.82679308790130689819020674812, −4.61977830716439079667819380804, −3.65307683645201894592301801425, −2.54006404426775800203284672436, −1.77361137302120758915139392589, −0.28056352961963526344828105281,
0.878242228840374217547190646074, 2.47680254987283298954188427455, 3.20975549532666909201892769790, 3.61276147282695002581874862634, 4.88399755543455127204067869784, 5.75219959449486048294857606000, 6.36325271423697887680751821787, 6.90440700637384824737151277184, 7.72541383604421495702031541456, 8.666438264618881848490867712512
