L(s) = 1 | + (−1.93 − 1.11i)5-s + (−2.26 − 3.92i)7-s + (12.9 − 7.46i)11-s + (−7.82 + 13.5i)13-s + 7.53i·17-s + 9.77·19-s + (−9.38 − 5.41i)23-s + (2.5 + 4.33i)25-s + (3.94 − 2.27i)29-s + (24.4 − 42.4i)31-s + 10.1i·35-s + 21.4·37-s + (45.6 + 26.3i)41-s + (−20.5 − 35.6i)43-s + (−32.2 + 18.6i)47-s + ⋯ |
L(s) = 1 | + (−0.387 − 0.223i)5-s + (−0.323 − 0.560i)7-s + (1.17 − 0.678i)11-s + (−0.602 + 1.04i)13-s + 0.443i·17-s + 0.514·19-s + (−0.408 − 0.235i)23-s + (0.100 + 0.173i)25-s + (0.136 − 0.0785i)29-s + (0.789 − 1.36i)31-s + 0.289i·35-s + 0.580·37-s + (1.11 + 0.642i)41-s + (−0.478 − 0.829i)43-s + (−0.685 + 0.395i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.409878747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409878747\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
good | 7 | \( 1 + (2.26 + 3.92i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-12.9 + 7.46i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (7.82 - 13.5i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 7.53iT - 289T^{2} \) |
| 19 | \( 1 - 9.77T + 361T^{2} \) |
| 23 | \( 1 + (9.38 + 5.41i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-3.94 + 2.27i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-24.4 + 42.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 21.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-45.6 - 26.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (20.5 + 35.6i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (32.2 - 18.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 81.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (70.3 + 40.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (24.5 + 42.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-23.2 + 40.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 17.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 101.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (28.7 + 49.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-75.2 + 43.4i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (90.4 + 156. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.155570592821691686521623984023, −8.160314949456473924578921178896, −7.43132798110101744187822598273, −6.52751727174616221570814920126, −5.95203433014204066359748933516, −4.52920159998597149152748515008, −4.07543057639064987942228575626, −3.03855618355321968351294598461, −1.63174080020162503762083608186, −0.43593448156658952356456738518,
1.11241837567071996716151303554, 2.53814067159643402082007436164, 3.35983029109675629571255952927, 4.42377608859260268311970779587, 5.30160261267680463820668943071, 6.25913803242374575189816762267, 7.04461926961041687046670034789, 7.76612320687610132540456975677, 8.656470004225360780615962650508, 9.505156427635028066952203487260