| L(s) = 1 | − 1.88i·5-s + 10.9·7-s − 0.171i·11-s + 6.48·13-s + 27.2i·17-s + 5.32·19-s + 3.51i·23-s + 21.4·25-s − 34.8i·29-s − 26.9·31-s − 20.6i·35-s + 46.4·37-s − 23.5i·41-s − 55.1·43-s + 57.5i·47-s + ⋯ |
| L(s) = 1 | − 0.376i·5-s + 1.56·7-s − 0.0155i·11-s + 0.498·13-s + 1.60i·17-s + 0.280·19-s + 0.152i·23-s + 0.858·25-s − 1.20i·29-s − 0.869·31-s − 0.590i·35-s + 1.25·37-s − 0.573i·41-s − 1.28·43-s + 1.22i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.381194893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.381194893\) |
| \(L(2)\) |
|
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 | \( 1 + 1.88iT - 25T^{2} \) |
| 7 | \( 1 - 10.9T + 49T^{2} \) |
| 11 | \( 1 + 0.171iT - 121T^{2} \) |
| 13 | \( 1 - 6.48T + 169T^{2} \) |
| 17 | \( 1 - 27.2iT - 289T^{2} \) |
| 19 | \( 1 - 5.32T + 361T^{2} \) |
| 23 | \( 1 - 3.51iT - 529T^{2} \) |
| 29 | \( 1 + 34.8iT - 841T^{2} \) |
| 31 | \( 1 + 26.9T + 961T^{2} \) |
| 37 | \( 1 - 46.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 57.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 82.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 79.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 66.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 115.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 36.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 158. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 62.8T + 9.40e3T^{2} \) |
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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
−10.01738907241699880327335986994, −8.943032977860955856760532365781, −8.216056693317977913035395139320, −7.71956808777231687534372253503, −6.41639792472371661823336492738, −5.48885147796703334697955546589, −4.61857266990813032871486211055, −3.73555773505185841239365025369, −2.10052713670322295955479425390, −1.12314034361555983269032075098,
1.03482434837598114471393726289, 2.30034285779999485815555919398, 3.51237806713281551766023317627, 4.83041614108548810716947378082, 5.29700690736686653854574077179, 6.68667215722502133570762333390, 7.41346973780739609738743596832, 8.279897470385875236782403234809, 9.018699732015682961369255012165, 10.02377579686807475161040359646
