L(s) = 1 | + (−2.50 − 4.34i)5-s + 7.91·7-s − 16.3·11-s + (−11.1 − 6.43i)13-s + (4.46 + 7.73i)17-s + (−18.5 + 4.20i)19-s + (−7.27 + 12.6i)23-s + (−0.0975 + 0.168i)25-s + (29.7 + 17.1i)29-s − 25.5i·31-s + (−19.8 − 34.4i)35-s + 49.2i·37-s + (−27.8 + 16.0i)41-s + (−4.51 − 7.81i)43-s + (−19.9 + 34.5i)47-s + ⋯ |
L(s) = 1 | + (−0.501 − 0.869i)5-s + 1.13·7-s − 1.48·11-s + (−0.857 − 0.495i)13-s + (0.262 + 0.455i)17-s + (−0.975 + 0.221i)19-s + (−0.316 + 0.548i)23-s + (−0.00390 + 0.00675i)25-s + (1.02 + 0.592i)29-s − 0.823i·31-s + (−0.567 − 0.983i)35-s + 1.33i·37-s + (−0.678 + 0.391i)41-s + (−0.104 − 0.181i)43-s + (−0.424 + 0.734i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04715583628\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04715583628\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (18.5 - 4.20i)T \) |
good | 5 | \( 1 + (2.50 + 4.34i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 - 7.91T + 49T^{2} \) |
| 11 | \( 1 + 16.3T + 121T^{2} \) |
| 13 | \( 1 + (11.1 + 6.43i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-4.46 - 7.73i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (7.27 - 12.6i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-29.7 - 17.1i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + 25.5iT - 961T^{2} \) |
| 37 | \( 1 - 49.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (27.8 - 16.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (4.51 + 7.81i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (19.9 - 34.5i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (73.6 + 42.5i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (53.9 - 31.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (38.1 - 66.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (69.8 + 40.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-11.0 + 6.37i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (42.2 + 73.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (106. - 61.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 21.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-98.0 - 56.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-146. + 84.5i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.958884056203416216053494465094, −8.596940440970461724598442184857, −8.083093147559372157467198205407, −7.54942173329740767755816675910, −6.03119908964820347312647932152, −4.88500610875259322596356229971, −4.61450191391323488578593091120, −2.96554909349977747170315550081, −1.60907196364124311943985958729, −0.01595219982601826374210418592,
2.08176004261845639375842515134, 3.01729640888822250729789702503, 4.49206924025103980242825779612, 5.13094189670109352634917998049, 6.47039209964549698220959460934, 7.46133569092836249034585789402, 7.943405101230829097888968115011, 8.915358678546615510863567546981, 10.24416711856514051850974120917, 10.68408357831635781864837068573