L(s) = 1 | + 3.16i·2-s − 2.00·4-s − 15i·7-s + 18.9i·8-s − 63.2·11-s + 35i·13-s + 47.4·14-s − 76·16-s + 88.5i·17-s − 91·19-s − 200i·22-s + 113. i·23-s − 110.·26-s + 30.0i·28-s − 63.2·29-s + ⋯ |
L(s) = 1 | + 1.11i·2-s − 0.250·4-s − 0.809i·7-s + 0.838i·8-s − 1.73·11-s + 0.746i·13-s + 0.905·14-s − 1.18·16-s + 1.26i·17-s − 1.09·19-s − 1.93i·22-s + 1.03i·23-s − 0.834·26-s + 0.202i·28-s − 0.404·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.187073 - 0.792454i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.187073 - 0.792454i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 3.16iT - 8T^{2} \) |
| 7 | \( 1 + 15iT - 343T^{2} \) |
| 11 | \( 1 + 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 88.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 91T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 63.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147T + 2.97e4T^{2} \) |
| 37 | \( 1 + 370iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 442.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 335iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 177. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 88.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 885.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 427T + 2.26e5T^{2} \) |
| 67 | \( 1 + 15iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 63.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 70iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 876T + 4.93e5T^{2} \) |
| 83 | \( 1 - 531. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3iT - 9.12e5T^{2} \) |
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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
−12.64616772570852353134770951958, −11.04711005982576152721063725798, −10.62514850568218668104245782257, −9.177668458964042154196522476501, −7.960969285221124169398637566560, −7.46135928998467572538510548669, −6.29775069127369547833828267176, −5.34509012550201650055277875612, −4.04624084963403353181420888668, −2.15565980484628573858913676225,
0.30324830918853772506236517044, 2.29879582618092112555991173880, 2.97943474939384084206824540315, 4.69474076164117674731632221597, 5.86598827803931735026584334301, 7.29784914026726195275685132207, 8.443434834850680752285119116856, 9.571936976051561855465745118854, 10.54635185491323010235097135491, 11.08878363009961207328711715637