| L(s) = 1 | + (−2.27 − 2.71i)3-s + (−5.68 − 2.07i)5-s + (−6.17 + 10.6i)7-s + (−0.612 + 3.47i)9-s + (−8.21 − 14.2i)11-s + (12.0 − 14.3i)13-s + (7.33 + 20.1i)15-s + (−0.283 − 1.60i)17-s + (−17.9 + 6.19i)19-s + (43.0 − 7.58i)21-s + (21.0 − 7.64i)23-s + (8.93 + 7.49i)25-s + (−16.7 + 9.68i)27-s + (13.0 + 2.30i)29-s + (−21.3 − 12.3i)31-s + ⋯ |
| L(s) = 1 | + (−0.758 − 0.903i)3-s + (−1.13 − 0.414i)5-s + (−0.881 + 1.52i)7-s + (−0.0681 + 0.386i)9-s + (−0.747 − 1.29i)11-s + (0.925 − 1.10i)13-s + (0.488 + 1.34i)15-s + (−0.0166 − 0.0944i)17-s + (−0.945 + 0.325i)19-s + (2.04 − 0.361i)21-s + (0.913 − 0.332i)23-s + (0.357 + 0.299i)25-s + (−0.621 + 0.358i)27-s + (0.450 + 0.0794i)29-s + (−0.687 − 0.397i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0308096 - 0.340076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0308096 - 0.340076i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (17.9 - 6.19i)T \) |
| good | 3 | \( 1 + (2.27 + 2.71i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (5.68 + 2.07i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (6.17 - 10.6i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (8.21 + 14.2i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-12.0 + 14.3i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (0.283 + 1.60i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (-21.0 + 7.64i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-13.0 - 2.30i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (21.3 + 12.3i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 29.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (10.7 + 12.8i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-19.3 - 7.03i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-3.01 + 17.0i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (13.7 + 37.7i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (86.1 - 15.1i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (18.8 - 6.86i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-42.6 - 7.52i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (4.32 - 11.8i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-12.7 + 10.7i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-13.3 - 15.8i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-61.6 + 106. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-58.3 + 69.5i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (111. - 19.6i)T + (8.84e3 - 3.21e3i)T^{2} \) |
| 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
−13.16012248679701476328091398746, −12.65304759515222004333188378915, −11.77830639504161402132811023387, −10.81182714461707260278516728245, −8.821957958816107746250950070499, −8.034179972651514587242108030757, −6.35524137650244042648399577340, −5.53034302171466341549139395456, −3.20293418935195132536861900551, −0.31116020875949310310964376921,
3.79763181909757508725535297562, 4.58113272577552984358315359000, 6.65854530794868883842974792953, 7.57271767799850132434745856941, 9.483063579398634353329402121192, 10.70240080057069408389731962160, 11.00654141622059888019160563428, 12.50732577145629301081091795036, 13.65794441667978373034217706859, 15.13173416592506994616199494526
