L(s) = 1 | − 5.71i·5-s − 5.91·7-s − 2.41i·11-s + 17.1·13-s + 16.1i·17-s − 4.35·19-s + 15.9i·23-s − 7.70·25-s − 31.2i·29-s + 13.5·31-s + 33.8i·35-s + 45.7·37-s − 30.3i·41-s + 21.0·43-s + 43.6i·47-s + ⋯ |
L(s) = 1 | − 1.14i·5-s − 0.845·7-s − 0.219i·11-s + 1.31·13-s + 0.949i·17-s − 0.229·19-s + 0.694i·23-s − 0.308·25-s − 1.07i·29-s + 0.435·31-s + 0.966i·35-s + 1.23·37-s − 0.740i·41-s + 0.490·43-s + 0.928i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.909792384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.909792384\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
good | 5 | \( 1 + 5.71iT - 25T^{2} \) |
| 7 | \( 1 + 5.91T + 49T^{2} \) |
| 11 | \( 1 + 2.41iT - 121T^{2} \) |
| 13 | \( 1 - 17.1T + 169T^{2} \) |
| 17 | \( 1 - 16.1iT - 289T^{2} \) |
| 23 | \( 1 - 15.9iT - 529T^{2} \) |
| 29 | \( 1 + 31.2iT - 841T^{2} \) |
| 31 | \( 1 - 13.5T + 961T^{2} \) |
| 37 | \( 1 - 45.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 21.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 43.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 43.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 86.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 112.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 41.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 13.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 21.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 71.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 19.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 20.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 20.1T + 9.40e3T^{2} \) |
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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
−8.500426966727475682033820876046, −8.056635334255227045032620629752, −6.94957503956220958935704101613, −5.94975666781687744526406280667, −5.74864020899212129521327473298, −4.39917885874753931996186538597, −3.90350850344438932354827495325, −2.84289634278180827298597191337, −1.51300042770659288329828796182, −0.63287509493955731255891897022,
0.802353819745480361231943411805, 2.29263458020024365362627951130, 3.11865558071565396404131421603, 3.73926694402559367294133655571, 4.84879343347608185279586714272, 5.91204019006235575173885202084, 6.67049672047000848787504261290, 6.91658160766410928144776389387, 8.015330566016224323962319163590, 8.752576527692228491183072279612