L(s) = 1 | + (4.24 + 3i)3-s − 5.65·5-s + 16.9i·7-s + (8.99 + 25.4i)9-s − 30i·11-s + 72i·13-s + (−24 − 16.9i)15-s − 50.9i·17-s + 25.4·19-s + (−50.9 + 71.9i)21-s − 144·23-s − 93·25-s + (−38.1 + 134. i)27-s − 5.65·29-s + 220. i·31-s + ⋯ |
L(s) = 1 | + (0.816 + 0.577i)3-s − 0.505·5-s + 0.916i·7-s + (0.333 + 0.942i)9-s − 0.822i·11-s + 1.53i·13-s + (−0.413 − 0.292i)15-s − 0.726i·17-s + 0.307·19-s + (−0.529 + 0.748i)21-s − 1.30·23-s − 0.743·25-s + (−0.272 + 0.962i)27-s − 0.0362·29-s + 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.556152407\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.556152407\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.24 - 3i)T \) |
good | 5 | \( 1 + 5.65T + 125T^{2} \) |
| 7 | \( 1 - 16.9iT - 343T^{2} \) |
| 11 | \( 1 + 30iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 72iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 50.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 25.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 144T + 1.21e4T^{2} \) |
| 29 | \( 1 + 5.65T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 72iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 305. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 576T + 1.03e5T^{2} \) |
| 53 | \( 1 - 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 414iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 504iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 789.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 720T + 3.57e5T^{2} \) |
| 73 | \( 1 + 178T + 3.89e5T^{2} \) |
| 79 | \( 1 - 967. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 438iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 865. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 650T + 9.12e5T^{2} \) |
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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
−11.54568142621854909639924261378, −10.18211744691432489061068908337, −9.301634422927926612318290574228, −8.631203660568514703233478403539, −7.82225400476871252646714895071, −6.58994381961976050526281556992, −5.29933565141488361177774835691, −4.19760928153514065926102985957, −3.17326869401691877885573219438, −1.94336330063024409060053066749,
0.45623467062911906826079450368, 1.95402057710093054964865053480, 3.42407954707002348390106755311, 4.22363940388934884837467127752, 5.83378996238198591387819113968, 7.07786769427252295565252297856, 7.80722387709199429916804167159, 8.334900435973290197739657412436, 9.803940147695615391795939227576, 10.28485734432653579003092119295