| L(s) = 1 | − 5.30i·3-s − 0.206i·7-s − 19.1·9-s − 15.0i·11-s + 11.6·13-s − 18.1·17-s − 19.3i·19-s − 1.09·21-s + 27.2i·23-s + 53.6i·27-s − 44.4·29-s − 20.3i·31-s − 79.6·33-s − 18.1·37-s − 62.0i·39-s + ⋯ |
| L(s) = 1 | − 1.76i·3-s − 0.0295i·7-s − 2.12·9-s − 1.36i·11-s + 0.899·13-s − 1.07·17-s − 1.02i·19-s − 0.0521·21-s + 1.18i·23-s + 1.98i·27-s − 1.53·29-s − 0.657i·31-s − 2.41·33-s − 0.489·37-s − 1.59i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9282284359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9282284359\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.30iT - 9T^{2} \) |
| 7 | \( 1 + 0.206iT - 49T^{2} \) |
| 11 | \( 1 + 15.0iT - 121T^{2} \) |
| 13 | \( 1 - 11.6T + 169T^{2} \) |
| 17 | \( 1 + 18.1T + 289T^{2} \) |
| 19 | \( 1 + 19.3iT - 361T^{2} \) |
| 23 | \( 1 - 27.2iT - 529T^{2} \) |
| 29 | \( 1 + 44.4T + 841T^{2} \) |
| 31 | \( 1 + 20.3iT - 961T^{2} \) |
| 37 | \( 1 + 18.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 32.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.06iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 5.37iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 79.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 83.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.51iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 41.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 15.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 50.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 10.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 12.1T + 9.40e3T^{2} \) |
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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
−9.069059731960662522350086989552, −8.655952534842980912567728847463, −7.68522188356356317420280028381, −6.98086725093210524287742367791, −6.12337922766096489935467386236, −5.49580163004286944527479668153, −3.74090575611522471794360290364, −2.60197153124634813652130622628, −1.44959180826445223743127088333, −0.30969205107723985077822090572,
2.10753362349802404947160607598, 3.54718742948212126577996930554, 4.24101782619381779988477186215, 5.02762416322072776691473402073, 5.99564867486937666347987574538, 7.11431142616212724809547440666, 8.455071220158467701745663638274, 8.992684942878293650935241953782, 9.892267846293928473176646350307, 10.44476685334445133294455682006
