| L(s) = 1 | + (1.59 + 1.59i)2-s + (1.22 − 1.22i)3-s + 1.11i·4-s + 3.91·6-s + (−1.87 − 1.87i)7-s + (4.61 − 4.61i)8-s − 2.99i·9-s − 13.7·11-s + (1.36 + 1.36i)12-s + (16.4 − 16.4i)13-s − 5.98i·14-s + 19.2·16-s + (3.05 + 3.05i)17-s + (4.79 − 4.79i)18-s − 4.66i·19-s + ⋯ |
| L(s) = 1 | + (0.799 + 0.799i)2-s + (0.408 − 0.408i)3-s + 0.278i·4-s + 0.652·6-s + (−0.267 − 0.267i)7-s + (0.576 − 0.576i)8-s − 0.333i·9-s − 1.24·11-s + (0.113 + 0.113i)12-s + (1.26 − 1.26i)13-s − 0.427i·14-s + 1.20·16-s + (0.179 + 0.179i)17-s + (0.266 − 0.266i)18-s − 0.245i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.963697058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.963697058\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.87 + 1.87i)T \) |
| good | 2 | \( 1 + (-1.59 - 1.59i)T + 4iT^{2} \) |
| 11 | \( 1 + 13.7T + 121T^{2} \) |
| 13 | \( 1 + (-16.4 + 16.4i)T - 169iT^{2} \) |
| 17 | \( 1 + (-3.05 - 3.05i)T + 289iT^{2} \) |
| 19 | \( 1 + 4.66iT - 361T^{2} \) |
| 23 | \( 1 + (-4.61 + 4.61i)T - 529iT^{2} \) |
| 29 | \( 1 + 50.3iT - 841T^{2} \) |
| 31 | \( 1 - 11.0T + 961T^{2} \) |
| 37 | \( 1 + (-44.4 - 44.4i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 20.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (41.9 - 41.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (20.4 + 20.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-46.1 + 46.1i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 47.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 33.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-63.1 - 63.1i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 31.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-19.0 + 19.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 53.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (97.3 - 97.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 156. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-76.2 - 76.2i)T + 9.40e3iT^{2} \) |
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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
−10.41756942539395878866075449974, −9.862475672862234811629312340222, −8.227540385900733944542193880630, −7.928462259175979840736332360109, −6.73390123347425263252729089216, −5.98105221809952216318633766223, −5.11312200903264592996246769224, −3.88787933279696123554478181774, −2.79061567783075979690906445099, −0.900125809997682764653632668979,
1.80798988574889711337118024451, 2.96147252702044491612957629061, 3.78790337122946837847382466768, 4.80436054496064905034482239659, 5.73685354044662354217452153477, 7.12262668517068643955818432746, 8.239333207031338624313483893273, 8.973902554208051618587776200650, 10.09628938356110926072500420228, 10.91800373975033232423734777250
