L(s) = 1 | + 1.73·3-s + 0.898i·5-s + 2.82i·7-s + 2.99·9-s + 4.38·11-s + 13.7i·13-s + 1.55i·15-s + 17.5·17-s − 4.38·19-s + 4.89i·21-s + 22.0i·23-s + 24.1·25-s + 5.19·27-s + 44.4i·29-s − 53.1i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.179i·5-s + 0.404i·7-s + 0.333·9-s + 0.398·11-s + 1.06i·13-s + 0.103i·15-s + 1.03·17-s − 0.230·19-s + 0.233i·21-s + 0.958i·23-s + 0.967·25-s + 0.192·27-s + 1.53i·29-s − 1.71i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.94437 + 0.805386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94437 + 0.805386i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
good | 5 | \( 1 - 0.898iT - 25T^{2} \) |
| 7 | \( 1 - 2.82iT - 49T^{2} \) |
| 11 | \( 1 - 4.38T + 121T^{2} \) |
| 13 | \( 1 - 13.7iT - 169T^{2} \) |
| 17 | \( 1 - 17.5T + 289T^{2} \) |
| 19 | \( 1 + 4.38T + 361T^{2} \) |
| 23 | \( 1 - 22.0iT - 529T^{2} \) |
| 29 | \( 1 - 44.4iT - 841T^{2} \) |
| 31 | \( 1 + 53.1iT - 961T^{2} \) |
| 37 | \( 1 - 35.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 49.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 1.70iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 24.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 93.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 123. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 131. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 110.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 73.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 105.T + 9.40e3T^{2} \) |
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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
−11.34909881943724613388055673417, −10.17129418913650425981160437013, −9.321707614761428162538304753609, −8.599774546783128876699540154102, −7.49631805892605710029205197048, −6.61265431177960003481889819111, −5.40231636281206399244207117109, −4.10892874441146498788447043225, −2.98666730153954462294220496686, −1.58005507792194573823468831628,
0.986987713126649011712094129468, 2.72498663524746244016189514844, 3.85158081887047015371004729627, 5.04042996098395006778367153140, 6.27964726279257400065906697744, 7.41875543435543240176239391153, 8.225395544302396386870417150214, 9.103469699898379597851136897941, 10.16576564020355425176624720161, 10.76115577921119155618535327300