| L(s) = 1 | + (−2.5 − 1.65i)3-s + (3.5 + 8.29i)9-s + 16.5i·11-s + 10·13-s + 3.31i·17-s − 7·19-s − 19.8i·23-s + (4.99 − 26.5i)27-s − 33.1i·29-s − 42·31-s + (27.5 − 41.4i)33-s − 40·37-s + (−25 − 16.5i)39-s + 16.5i·41-s + 50·43-s + ⋯ |
| L(s) = 1 | + (−0.833 − 0.552i)3-s + (0.388 + 0.921i)9-s + 1.50i·11-s + 0.769·13-s + 0.195i·17-s − 0.368·19-s − 0.865i·23-s + (0.185 − 0.982i)27-s − 1.14i·29-s − 1.35·31-s + (0.833 − 1.25i)33-s − 1.08·37-s + (−0.641 − 0.425i)39-s + 0.404i·41-s + 1.16·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3148429852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3148429852\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.5 + 1.65i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 16.5iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 - 3.31iT - 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 + 19.8iT - 529T^{2} \) |
| 29 | \( 1 + 33.1iT - 841T^{2} \) |
| 31 | \( 1 + 42T + 961T^{2} \) |
| 37 | \( 1 + 40T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 46.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 45T + 4.48e3T^{2} \) |
| 71 | \( 1 - 33.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 35T + 5.32e3T^{2} \) |
| 79 | \( 1 + 12T + 6.24e3T^{2} \) |
| 83 | \( 1 + 69.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 149. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 70T + 9.40e3T^{2} \) |
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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
−10.06080987981484462435677213920, −9.104046073387425838131259919499, −8.070271355550119594556357116038, −7.31657121709517981562226636831, −6.56371296774141269459060328706, −5.80617628543334828309934118988, −4.79452945446858682497260788802, −4.02521801472217760338777038469, −2.37602536304731775799188247711, −1.42142824367537163018304147781,
0.10996348351007812901318932232, 1.40654623691832487618054811898, 3.26425557286512377540173694717, 3.86249104009996501345695392145, 5.14866743377138493509559222844, 5.75075852927729883882626936139, 6.51660632905670742941123477004, 7.49069277738582295931583928480, 8.709639531485777815012602388886, 9.104840705362244004613420630956
