L(s) = 1 | + (8.10 + 7.70i)5-s + 22.2i·7-s − 1.79·11-s + 58.2i·13-s + 18.9i·17-s + 104.·19-s − 49.6i·23-s + (6.37 + 124. i)25-s − 293.·29-s − 64.4·31-s + (−171. + 180i)35-s + 19.8i·37-s + 165.·41-s + 247. i·43-s − 384. i·47-s + ⋯ |
L(s) = 1 | + (0.724 + 0.688i)5-s + 1.19i·7-s − 0.0490·11-s + 1.24i·13-s + 0.270i·17-s + 1.26·19-s − 0.449i·23-s + (0.0509 + 0.998i)25-s − 1.87·29-s − 0.373·31-s + (−0.826 + 0.869i)35-s + 0.0883i·37-s + 0.630·41-s + 0.877i·43-s − 1.19i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 - 0.688i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.724 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.848439930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.848439930\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-8.10 - 7.70i)T \) |
good | 7 | \( 1 - 22.2iT - 343T^{2} \) |
| 11 | \( 1 + 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.8iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 247. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 384. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 463. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 73.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 173. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 320. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 173. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 384. iT - 9.12e5T^{2} \) |
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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
−10.22685917584071793725426803812, −9.363780269095593972204098598336, −8.936175134320503544711572672320, −7.64580665282854810885259455439, −6.74175498064494535197159883119, −5.87709887187600333241532550495, −5.16854727168092994149537940102, −3.71213817882796943753970028791, −2.54704828842364215306893214284, −1.69981276230065148776252260041,
0.50083597825191823173676834139, 1.50028072794779841463571661946, 3.04933660917032998174598126845, 4.15306207506281699291302613090, 5.28200316590202535669379322091, 5.89330648341779241396589488719, 7.32760523999665518595100334166, 7.73609341922517939699695810462, 9.011724749245625832574039782162, 9.695684310981481212772190774882