| L(s) = 1 | + 4.18·5-s − 2.58i·7-s − 6.38i·11-s − 11.5·13-s − 7.58·17-s + 0.0311i·19-s + 7.52i·23-s − 7.52·25-s − 13.5·29-s − 29.7i·31-s − 10.8i·35-s − 57.4·37-s − 27.1·41-s + 77.8i·43-s − 43.8i·47-s + ⋯ |
| L(s) = 1 | + 0.836·5-s − 0.369i·7-s − 0.580i·11-s − 0.891·13-s − 0.446·17-s + 0.00163i·19-s + 0.327i·23-s − 0.300·25-s − 0.467·29-s − 0.960i·31-s − 0.308i·35-s − 1.55·37-s − 0.662·41-s + 1.81i·43-s − 0.932i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2716553357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2716553357\) |
| \(L(2)\) |
|
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 | \( 1 - 4.18T + 25T^{2} \) |
| 7 | \( 1 + 2.58iT - 49T^{2} \) |
| 11 | \( 1 + 6.38iT - 121T^{2} \) |
| 13 | \( 1 + 11.5T + 169T^{2} \) |
| 17 | \( 1 + 7.58T + 289T^{2} \) |
| 19 | \( 1 - 0.0311iT - 361T^{2} \) |
| 23 | \( 1 - 7.52iT - 529T^{2} \) |
| 29 | \( 1 + 13.5T + 841T^{2} \) |
| 31 | \( 1 + 29.7iT - 961T^{2} \) |
| 37 | \( 1 + 57.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 27.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 77.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 43.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 87.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 67.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 23.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 67.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 8.55iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 5.22T + 7.92e3T^{2} \) |
| 97 | \( 1 - 120.T + 9.40e3T^{2} \) |
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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
−8.855620785419424707267951235997, −7.914120094801324085468043680706, −7.14026678228488951887816228005, −6.27101843140572967820944420536, −5.53476033108673794894234697556, −4.67887927532507045512817348426, −3.61508117284610578767249319534, −2.53320683191847750127347708194, −1.54693129803942924866378478296, −0.06387286715038366549294532381,
1.72018799968978064860432745360, 2.40090645648639848557854998537, 3.59407952345187560990045478610, 4.83730160867653260588108901439, 5.35714111865547318899085548350, 6.38770557107985856133370248969, 7.04369765728464172475107914349, 7.949792565600019890408335397911, 8.950112342889067133629983135405, 9.455857874630556784314606597659
