| L(s) = 1 | + (−4 + 3i)5-s − 6i·7-s − 21i·11-s − 15i·13-s − 23·17-s − 14·19-s − 7·23-s + (7 − 24i)25-s + 3i·29-s + 25·31-s + (18 + 24i)35-s + 54i·37-s + 24i·41-s − 15i·43-s − 49·47-s + ⋯ |
| L(s) = 1 | + (−0.800 + 0.600i)5-s − 0.857i·7-s − 1.90i·11-s − 1.15i·13-s − 1.35·17-s − 0.736·19-s − 0.304·23-s + (0.280 − 0.959i)25-s + 0.103i·29-s + 0.806·31-s + (0.514 + 0.685i)35-s + 1.45i·37-s + 0.585i·41-s − 0.348i·43-s − 1.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 - 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.599 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04694623413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04694623413\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4 - 3i)T \) |
| good | 7 | \( 1 + 6iT - 49T^{2} \) |
| 11 | \( 1 + 21iT - 121T^{2} \) |
| 13 | \( 1 + 15iT - 169T^{2} \) |
| 17 | \( 1 + 23T + 289T^{2} \) |
| 19 | \( 1 + 14T + 361T^{2} \) |
| 23 | \( 1 + 7T + 529T^{2} \) |
| 29 | \( 1 - 3iT - 841T^{2} \) |
| 31 | \( 1 - 25T + 961T^{2} \) |
| 37 | \( 1 - 54iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 24iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49T + 2.20e3T^{2} \) |
| 53 | \( 1 + 14T + 2.80e3T^{2} \) |
| 59 | \( 1 - 30iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 44T + 3.72e3T^{2} \) |
| 67 | \( 1 + 66iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 18iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 37T + 6.24e3T^{2} \) |
| 83 | \( 1 - 116T + 6.88e3T^{2} \) |
| 89 | \( 1 - 126iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 78iT - 9.40e3T^{2} \) |
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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
−8.139332023431971342651781446104, −7.912795898603248626650101613621, −6.60706329420804462687064696058, −6.37021739616867946298515004416, −5.12319594817142782028272555004, −4.12963804115836870454264958039, −3.40691268195213294238785353882, −2.65102036837822301596444776260, −0.887468530353090504074733597837, −0.01422834799520005410633619407,
1.79231667592502237644855025500, 2.39838282945769459890363604942, 4.04406245595753141545892912866, 4.44309434217422769501123118867, 5.22208746320944278373890869042, 6.45626570471253621614814826486, 7.03106935979234563809439921461, 7.88572883373403207945809516205, 8.773824679372388873577178650712, 9.191118487247850138397578281019
