| L(s) = 1 | + 6.42·5-s − 13.8i·7-s + 13.0i·11-s − 7.16·13-s − 31.5·17-s − 16.4i·19-s + 16.9i·23-s + 16.2·25-s − 42.4·29-s − 29.6i·31-s − 88.6i·35-s − 39.3·37-s − 39.8·41-s − 16.3i·43-s + 57.8i·47-s + ⋯ |
| L(s) = 1 | + 1.28·5-s − 1.97i·7-s + 1.18i·11-s − 0.551·13-s − 1.85·17-s − 0.867i·19-s + 0.738i·23-s + 0.649·25-s − 1.46·29-s − 0.957i·31-s − 2.53i·35-s − 1.06·37-s − 0.972·41-s − 0.379i·43-s + 1.23i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6193585150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6193585150\) |
| \(L(2)\) |
|
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 | \( 1 - 6.42T + 25T^{2} \) |
| 7 | \( 1 + 13.8iT - 49T^{2} \) |
| 11 | \( 1 - 13.0iT - 121T^{2} \) |
| 13 | \( 1 + 7.16T + 169T^{2} \) |
| 17 | \( 1 + 31.5T + 289T^{2} \) |
| 19 | \( 1 + 16.4iT - 361T^{2} \) |
| 23 | \( 1 - 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 42.4T + 841T^{2} \) |
| 31 | \( 1 + 29.6iT - 961T^{2} \) |
| 37 | \( 1 + 39.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 16.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 57.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 14.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 32.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 22.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 61.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 44.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 1.95T + 7.92e3T^{2} \) |
| 97 | \( 1 + 44.5T + 9.40e3T^{2} \) |
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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
−9.016222661808866000726368725147, −7.69475016117251837969492493558, −7.00606655941636295797215842773, −6.64452225423582194471106752747, −5.36827927898436798751516587354, −4.55876020095459779505628665476, −3.85463717938854934948957423168, −2.34782539293609483553473203083, −1.60117924682558629388917949967, −0.13973506312835965792146059153,
1.90841552446221222199871159667, 2.32464003110700278898184809727, 3.41803066173297462216340164345, 4.96429657554896713750390127861, 5.57252903610876299402444494939, 6.14767410736987730532355193124, 6.86459740120691588839234092660, 8.410602945224874756276955081239, 8.741300101351661723782722235281, 9.352510811980193662609214746210
