L(s) = 1 | + 0.445i·2-s + 1.80·4-s − 0.246i·5-s + 1.75i·7-s + 1.69i·8-s + 0.109·10-s − 5.65i·11-s − 0.780·14-s + 2.85·16-s − 3.80·17-s − 5.58i·19-s − 0.445i·20-s + 2.51·22-s + 8.34·23-s + 4.93·25-s + ⋯ |
L(s) = 1 | + 0.314i·2-s + 0.900·4-s − 0.110i·5-s + 0.662i·7-s + 0.598i·8-s + 0.0347·10-s − 1.70i·11-s − 0.208·14-s + 0.712·16-s − 0.922·17-s − 1.28i·19-s − 0.0995i·20-s + 0.536·22-s + 1.74·23-s + 0.987·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.207072551\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.207072551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.445iT - 2T^{2} \) |
| 5 | \( 1 + 0.246iT - 5T^{2} \) |
| 7 | \( 1 - 1.75iT - 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 + 5.58iT - 19T^{2} \) |
| 23 | \( 1 - 8.34T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 5.26iT - 31T^{2} \) |
| 37 | \( 1 - 3.19iT - 37T^{2} \) |
| 41 | \( 1 - 0.445iT - 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 - 6.73iT - 47T^{2} \) |
| 53 | \( 1 - 1.06T + 53T^{2} \) |
| 59 | \( 1 - 13.7iT - 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 + 5.96iT - 67T^{2} \) |
| 71 | \( 1 + 5.71iT - 71T^{2} \) |
| 73 | \( 1 - 7.35iT - 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 0.137iT - 89T^{2} \) |
| 97 | \( 1 - 13.6iT - 97T^{2} \) |
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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
−9.001570985845364020785127966421, −8.844892250592373387560143445329, −7.88045696099171752024499573259, −6.86095582939961543880537460676, −6.33208862822294755377789555931, −5.49671165010012773147587138010, −4.64062486260569533061104556448, −3.03239759502229856393569352544, −2.64356052616917319794718643806, −0.992494258358240918918761388583,
1.27193975869277575021273776440, 2.26973820425414383438607037656, 3.30924520687169110370847047571, 4.35488474677051066076137446468, 5.20219081832251818454859301890, 6.62775814183672384332598835829, 6.90717837923794160994978228313, 7.63929466207289476574196420897, 8.706293473742422021475113145093, 9.690668421265834334970914295879