| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 7.74i·5-s − 2.64·7-s − 2.82i·8-s − 10.9·10-s − 11.5i·11-s + 19.3·13-s − 3.74i·14-s + 4.00·16-s − 18.8i·17-s − 18.8·19-s − 15.4i·20-s + 16.3·22-s − 30.2i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.54i·5-s − 0.377·7-s − 0.353i·8-s − 1.09·10-s − 1.05i·11-s + 1.48·13-s − 0.267i·14-s + 0.250·16-s − 1.10i·17-s − 0.993·19-s − 0.774i·20-s + 0.742·22-s − 1.31i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.352032757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.352032757\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 7.74iT - 25T^{2} \) |
| 11 | \( 1 + 11.5iT - 121T^{2} \) |
| 13 | \( 1 - 19.3T + 169T^{2} \) |
| 17 | \( 1 + 18.8iT - 289T^{2} \) |
| 19 | \( 1 + 18.8T + 361T^{2} \) |
| 23 | \( 1 + 30.2iT - 529T^{2} \) |
| 29 | \( 1 + 57.1iT - 841T^{2} \) |
| 31 | \( 1 + 10.7T + 961T^{2} \) |
| 37 | \( 1 - 6.77T + 1.36e3T^{2} \) |
| 41 | \( 1 - 60.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 19.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 9.68iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 58.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 43.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 36.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 45.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 56.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 52.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 14.1T + 9.40e3T^{2} \) |
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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
−9.613491796047890103758740869686, −8.595396395433478510273434718707, −7.976086445697253898758723327578, −6.87914579376690461696172232974, −6.33616194079809563247901153450, −5.83636434373156405715505202730, −4.31057428901511683105825186865, −3.39915240835456237798966955332, −2.52160415731364413516051089594, −0.46093420008804662228357391198,
1.16174865354588852862723110683, 1.88147572822153845325244645671, 3.60127453340536929137709336584, 4.21715876110363692701787341356, 5.22280602267146786174497397970, 6.00257665425032764506685802212, 7.24484448427280826908407179758, 8.433528695578209437454991995534, 8.813236101491145748056890214487, 9.536508676627197374495282103642
