| 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 | \( 1 + (111. + 19.6i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−15.13173416592506994616199494526, −13.65794441667978373034217706859, −12.50732577145629301081091795036, −11.00654141622059888019160563428, −10.70240080057069408389731962160, −9.483063579398634353329402121192, −7.57271767799850132434745856941, −6.65854530794868883842974792953, −4.58113272577552984358315359000, −3.79763181909757508725535297562,
0.31116020875949310310964376921, 3.20293418935195132536861900551, 5.53034302171466341549139395456, 6.35524137650244042648399577340, 8.034179972651514587242108030757, 8.821957958816107746250950070499, 10.81182714461707260278516728245, 11.77830639504161402132811023387, 12.65304759515222004333188378915, 13.16012248679701476328091398746
