L(s) = 1 | − 2.79·2-s − 1.73i·3-s + 3.79·4-s + 2.23i·5-s + 4.83i·6-s + (4.15 − 5.63i)7-s + 0.578·8-s − 2.99·9-s − 6.24i·10-s − 18.9·11-s − 6.56i·12-s − 10.9i·13-s + (−11.6 + 15.7i)14-s + 3.87·15-s − 16.7·16-s − 22.3i·17-s + ⋯ |
L(s) = 1 | − 1.39·2-s − 0.577i·3-s + 0.948·4-s + 0.447i·5-s + 0.805i·6-s + (0.593 − 0.804i)7-s + 0.0723·8-s − 0.333·9-s − 0.624i·10-s − 1.72·11-s − 0.547i·12-s − 0.844i·13-s + (−0.829 + 1.12i)14-s + 0.258·15-s − 1.04·16-s − 1.31i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.202606 - 0.401437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.202606 - 0.401437i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (-4.15 + 5.63i)T \) |
good | 2 | \( 1 + 2.79T + 4T^{2} \) |
| 11 | \( 1 + 18.9T + 121T^{2} \) |
| 13 | \( 1 + 10.9iT - 169T^{2} \) |
| 17 | \( 1 + 22.3iT - 289T^{2} \) |
| 19 | \( 1 + 19.6iT - 361T^{2} \) |
| 23 | \( 1 + 31.9T + 529T^{2} \) |
| 29 | \( 1 - 39.9T + 841T^{2} \) |
| 31 | \( 1 - 36.6iT - 961T^{2} \) |
| 37 | \( 1 - 8.94T + 1.36e3T^{2} \) |
| 41 | \( 1 + 37.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 18.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 49.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 35.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 21.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 36.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 6.66iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 16.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 36.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 88.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 133. iT - 9.40e3T^{2} \) |
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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
−13.33259629105202563913402623902, −11.76900761146268282593532108926, −10.59703106163217013093933193170, −10.17914488186298381058474710026, −8.544408372490127519899916334328, −7.70711336064291108466150605166, −7.01750396474060923988551921446, −5.06005201179218654675815240708, −2.54400904959764824688823624802, −0.51063846384969029906319308001,
2.06014887344532381242191703502, 4.52330250120004627104984218749, 5.93908391441143012209557353923, 8.092952931459838735743323723077, 8.256411030146145078230295384896, 9.634855013615896950451006728853, 10.36990252454017672188159090718, 11.41999917964680342122604275450, 12.62082075552412121263511790913, 14.01841470292065194139303389041