L(s) = 1 | + (1.16 − 0.673i)3-s + (0.0412 − 2.64i)7-s + (−0.592 + 1.02i)9-s + (−0.705 − 1.22i)11-s − 6.47i·13-s + (−2.24 + 1.29i)17-s + (0.581 − 1.00i)19-s + (−1.73 − 3.11i)21-s + (1.36 + 0.786i)23-s + 5.63i·27-s − 6.22·29-s + (−2.49 − 4.31i)31-s + (−1.64 − 0.950i)33-s + (−7.34 − 4.23i)37-s + (−4.36 − 7.55i)39-s + ⋯ |
L(s) = 1 | + (0.673 − 0.388i)3-s + (0.0155 − 0.999i)7-s + (−0.197 + 0.342i)9-s + (−0.212 − 0.368i)11-s − 1.79i·13-s + (−0.545 + 0.314i)17-s + (0.133 − 0.230i)19-s + (−0.378 − 0.679i)21-s + (0.284 + 0.164i)23-s + 1.08i·27-s − 1.15·29-s + (−0.447 − 0.775i)31-s + (−0.286 − 0.165i)33-s + (−1.20 − 0.696i)37-s + (−0.698 − 1.20i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.560358725\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.560358725\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.0412 + 2.64i)T \) |
good | 3 | \( 1 + (-1.16 + 0.673i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.705 + 1.22i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.47iT - 13T^{2} \) |
| 17 | \( 1 + (2.24 - 1.29i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.581 + 1.00i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.36 - 0.786i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.22T + 29T^{2} \) |
| 31 | \( 1 + (2.49 + 4.31i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.34 + 4.23i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 + 7.24iT - 43T^{2} \) |
| 47 | \( 1 + (-9.47 - 5.47i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.0 + 6.39i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.02 + 1.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.307 + 0.533i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.26 - 3.04i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 + (-3.04 + 1.75i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.439 + 0.761i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.90iT - 83T^{2} \) |
| 89 | \( 1 + (-2.98 + 5.16i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.23iT - 97T^{2} \) |
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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
−9.085017824673732577495895842608, −8.443942885718409023625090626946, −7.46056952104624277464712295327, −7.37649137854558336551311417016, −5.89460768302435313702219626734, −5.20739304195435861168161853213, −3.91404154478595979488170722237, −3.12462284704342374013450459849, −2.05294492260353163343571560232, −0.55088525838216558279183907065,
1.84671688051961534281591628502, 2.72411470113670394356475524805, 3.80701928443936039400819719865, 4.66084897142596969571465652764, 5.68142049654243898022097419796, 6.61297807105289287907264676184, 7.39717172268345108322107416823, 8.612286445215765223015979355557, 9.027166135385144844986738937041, 9.486281815885743924154312898517