| L(s) = 1 | + (−0.5 + 1.86i)2-s + (−1.36 − 2.36i)3-s + (0.232 + 0.133i)4-s + (−4.36 − 4.36i)5-s + (5.09 − 1.36i)6-s + (2.26 + 8.46i)7-s + (−5.83 + 5.83i)8-s + (0.767 − 1.33i)9-s + (10.3 − 5.96i)10-s + (6.19 + 1.66i)11-s − 0.732i·12-s + (−6.5 − 11.2i)13-s − 16.9·14-s + (−4.36 + 16.2i)15-s + (−7.42 − 12.8i)16-s + (9.99 + 5.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.250 + 0.933i)2-s + (−0.455 − 0.788i)3-s + (0.0580 + 0.0334i)4-s + (−0.873 − 0.873i)5-s + (0.849 − 0.227i)6-s + (0.323 + 1.20i)7-s + (−0.728 + 0.728i)8-s + (0.0853 − 0.147i)9-s + (1.03 − 0.596i)10-s + (0.563 + 0.150i)11-s − 0.0610i·12-s + (−0.5 − 0.866i)13-s − 1.20·14-s + (−0.291 + 1.08i)15-s + (−0.464 − 0.804i)16-s + (0.587 + 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.586286 + 0.169291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586286 + 0.169291i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (6.5 + 11.2i)T \) |
| good | 2 | \( 1 + (0.5 - 1.86i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (1.36 + 2.36i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (4.36 + 4.36i)T + 25iT^{2} \) |
| 7 | \( 1 + (-2.26 - 8.46i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-6.19 - 1.66i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-9.99 - 5.76i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-3.36 + 0.901i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (8.49 - 4.90i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-5.69 - 9.86i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-1.92 - 1.92i)T + 961iT^{2} \) |
| 37 | \( 1 + (42.1 + 11.2i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-5.08 + 18.9i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-45 - 25.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-0.320 + 0.320i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 78.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-10.9 - 40.9i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (49.1 - 85.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-19.9 + 74.5i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (31.0 - 8.31i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-48.2 + 48.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 82.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (69.5 + 69.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (31.8 + 8.52i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-74.8 + 20.0i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−19.69946131770991524875045309811, −18.20982450485576413706454723373, −17.22051251081396437134079460275, −15.88555550706980427605586348646, −14.92450749565529743467736844708, −12.35508748671146947038322297182, −11.94096394180814851765994738485, −8.748529320555201240166277877670, −7.49426845231199612039305488387, −5.69553781794938833798720680035,
3.95683416420677932391690554448, 7.14798786949140281813477201047, 9.949203577202307198335670126866, 10.93174628761151014090250123475, 11.79909410586806601335509550978, 14.23948615269701998674708152208, 15.69349632537767292893142502756, 16.93927975372165790455416367734, 18.79531553297321548176777089468, 19.67380957328236826127553915476
