| L(s) = 1 | + (−1.70 + 1.23i)3-s + (2.48 + 7.65i)5-s + (1.55 − 2.14i)7-s + (−1.41 + 4.35i)9-s + (9.10 + 6.17i)11-s + (−21.4 − 6.95i)13-s + (−13.6 − 9.95i)15-s + (16.8 − 5.47i)17-s + (4.10 + 5.64i)19-s + 5.56i·21-s − 18.8·23-s + (−32.1 + 23.3i)25-s + (−8.82 − 27.1i)27-s + (−31.6 + 43.6i)29-s + (−3.81 + 11.7i)31-s + ⋯ |
| L(s) = 1 | + (−0.567 + 0.412i)3-s + (0.497 + 1.53i)5-s + (0.222 − 0.305i)7-s + (−0.157 + 0.483i)9-s + (0.827 + 0.561i)11-s + (−1.64 − 0.535i)13-s + (−0.913 − 0.663i)15-s + (0.991 − 0.322i)17-s + (0.215 + 0.297i)19-s + 0.264i·21-s − 0.818·23-s + (−1.28 + 0.935i)25-s + (−0.326 − 1.00i)27-s + (−1.09 + 1.50i)29-s + (−0.123 + 0.378i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.338830 + 1.06079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.338830 + 1.06079i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.55 + 2.14i)T \) |
| 11 | \( 1 + (-9.10 - 6.17i)T \) |
| good | 3 | \( 1 + (1.70 - 1.23i)T + (2.78 - 8.55i)T^{2} \) |
| 5 | \( 1 + (-2.48 - 7.65i)T + (-20.2 + 14.6i)T^{2} \) |
| 13 | \( 1 + (21.4 + 6.95i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (-16.8 + 5.47i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-4.10 - 5.64i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + 18.8T + 529T^{2} \) |
| 29 | \( 1 + (31.6 - 43.6i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (3.81 - 11.7i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (53.6 + 38.9i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-17.0 - 23.4i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 73.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-60.6 + 44.0i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (11.1 - 34.3i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-37.0 - 26.8i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-33.8 + 10.9i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 1.65T + 4.48e3T^{2} \) |
| 71 | \( 1 + (27.5 + 84.8i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-43.4 + 59.7i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-59.1 - 19.2i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-42.5 + 13.8i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + 104.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-20.1 + 62.0i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.70805642407608412390087336342, −10.70093928783376744856676650250, −10.21250814833417804685412958291, −9.455328609566194459226872253786, −7.62293136686939726876386492731, −7.12159617239099523619348768076, −5.86526256407232928492952297080, −4.95209211872882023063369283625, −3.45784195650463018979286621276, −2.13802776145015600706267249765,
0.56372680378382588638872031603, 1.91338486655549134634426277569, 4.01448186152504366754189807928, 5.30039951750786782582375184642, 5.87163657966411787695375442146, 7.15718429326045714232537364609, 8.380482388282514708863128666724, 9.273097638091633338244855231688, 9.879635418915613645397407562697, 11.56385785990193289047412944503
