L(s) = 1 | + 2.97·3-s − 4.37i·5-s + i·7-s + 5.83·9-s − i·11-s + (1.31 − 3.35i)13-s − 13.0i·15-s − 5.05·17-s − 0.365i·19-s + 2.97i·21-s − 3.66·23-s − 14.1·25-s + 8.42·27-s + 8.10·29-s + 7.09i·31-s + ⋯ |
L(s) = 1 | + 1.71·3-s − 1.95i·5-s + 0.377i·7-s + 1.94·9-s − 0.301i·11-s + (0.363 − 0.931i)13-s − 3.35i·15-s − 1.22·17-s − 0.0837i·19-s + 0.648i·21-s − 0.764·23-s − 2.83·25-s + 1.62·27-s + 1.50·29-s + 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.345485232\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.345485232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-1.31 + 3.35i)T \) |
good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 + 4.37iT - 5T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 0.365iT - 19T^{2} \) |
| 23 | \( 1 + 3.66T + 23T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 - 7.09iT - 31T^{2} \) |
| 37 | \( 1 + 6.97iT - 37T^{2} \) |
| 41 | \( 1 + 3.30iT - 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 8.81T + 53T^{2} \) |
| 59 | \( 1 - 8.55iT - 59T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 + 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 5.58iT - 71T^{2} \) |
| 73 | \( 1 - 1.87iT - 73T^{2} \) |
| 79 | \( 1 - 0.669T + 79T^{2} \) |
| 83 | \( 1 - 12.6iT - 83T^{2} \) |
| 89 | \( 1 + 8.69iT - 89T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−8.359628539128331121229975857352, −7.990704439885088510031553728822, −6.98757459471068740167843300031, −5.84540508056115911319870681193, −5.07058000835337969461411252541, −4.28094990469653159433985372525, −3.64498234548286335188109567110, −2.56184328562251917131965527569, −1.79018538501052847042082728509, −0.71083394380906275446218785625,
1.76760671507219167985140836538, 2.52981334937658678264570800402, 3.00068654439985516242372384182, 4.04560355747587710233071472158, 4.30574625554738720463631381136, 6.16362878789557687161403012839, 6.68379171961044124035731675353, 7.25431453377739711151665350864, 7.932151796631120002624556624377, 8.539642020388468241891807680003