| L(s) = 1 | + 6.08·5-s + 8.85i·7-s − 6.64i·11-s − 2.95·13-s − 21.7·17-s + 4.35i·19-s + 30.4i·23-s + 12.0·25-s + 17.0·29-s − 58.0i·31-s + 53.8i·35-s − 44.3·37-s + 5.33·41-s + 50.1i·43-s + 74.0i·47-s + ⋯ |
| L(s) = 1 | + 1.21·5-s + 1.26i·7-s − 0.603i·11-s − 0.227·13-s − 1.28·17-s + 0.229i·19-s + 1.32i·23-s + 0.481·25-s + 0.586·29-s − 1.87i·31-s + 1.53i·35-s − 1.19·37-s + 0.130·41-s + 1.16i·43-s + 1.57i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.186073772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186073772\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 - 6.08T + 25T^{2} \) |
| 7 | \( 1 - 8.85iT - 49T^{2} \) |
| 11 | \( 1 + 6.64iT - 121T^{2} \) |
| 13 | \( 1 + 2.95T + 169T^{2} \) |
| 17 | \( 1 + 21.7T + 289T^{2} \) |
| 23 | \( 1 - 30.4iT - 529T^{2} \) |
| 29 | \( 1 - 17.0T + 841T^{2} \) |
| 31 | \( 1 + 58.0iT - 961T^{2} \) |
| 37 | \( 1 + 44.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 5.33T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 74.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 18.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 31.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 74.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 50.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.81iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 25.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 3.61T + 7.92e3T^{2} \) |
| 97 | \( 1 - 140.T + 9.40e3T^{2} \) |
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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
−9.191566138956038445846259128072, −8.354390686410321687974392778437, −7.49491510551488605904878099071, −6.29948349333300215437494021026, −5.99222240707095220501156191225, −5.29756461846311538504160595030, −4.35630686791707573085826520867, −3.03694844947121546748223530297, −2.32180423382941670360975069803, −1.49757327837425730941318960662,
0.24108830585436803328977178994, 1.55623778703677345027567789669, 2.32716935548829921573177326141, 3.47965904749665494263490555903, 4.59379635740901272905802103058, 5.00238223725183235253728908871, 6.23710235007459665239126857538, 6.80686052623900313708245075432, 7.31880492415106499890038261788, 8.528849861365846103968429849455
