L(s) = 1 | − 3.41i·3-s − 9.09·5-s − 4.38·7-s − 2.63·9-s + 18.2·11-s + 11.8i·13-s + 31.0i·15-s − 8.65·17-s + 14.9i·21-s + 33.6·23-s + 57.6·25-s − 21.7i·27-s − 49.6i·29-s + 9.75i·31-s − 62.2i·33-s + ⋯ |
L(s) = 1 | − 1.13i·3-s − 1.81·5-s − 0.626·7-s − 0.292·9-s + 1.65·11-s + 0.912i·13-s + 2.06i·15-s − 0.509·17-s + 0.712i·21-s + 1.46·23-s + 2.30·25-s − 0.804i·27-s − 1.71i·29-s + 0.314i·31-s − 1.88i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.078498014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078498014\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 3.41iT - 9T^{2} \) |
| 5 | \( 1 + 9.09T + 25T^{2} \) |
| 7 | \( 1 + 4.38T + 49T^{2} \) |
| 11 | \( 1 - 18.2T + 121T^{2} \) |
| 13 | \( 1 - 11.8iT - 169T^{2} \) |
| 17 | \( 1 + 8.65T + 289T^{2} \) |
| 23 | \( 1 - 33.6T + 529T^{2} \) |
| 29 | \( 1 + 49.6iT - 841T^{2} \) |
| 31 | \( 1 - 9.75iT - 961T^{2} \) |
| 37 | \( 1 - 44.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 43.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 5.45T + 1.84e3T^{2} \) |
| 47 | \( 1 - 15.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 16.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 37.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 8.06iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 123. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.12T + 5.32e3T^{2} \) |
| 79 | \( 1 + 77.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 61.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 33.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 188. iT - 9.40e3T^{2} \) |
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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
−8.842173424270564105799938712306, −8.204360849056542577144089461855, −7.08258469261931282469497721186, −7.00711418412824492287170248996, −6.18889971572735593977963286681, −4.47438387217115703503291137042, −3.99700030755753326730923872560, −2.93021092756826490493590045332, −1.45188257768638788167954893104, −0.42503368577258386588701547114,
0.939810900782701365963715083193, 3.22302981245822281366023028315, 3.58639840543684735375292012215, 4.39502087676144708715143325156, 5.11585612843853114450648529056, 6.55400441418926520665117363395, 7.15850542562566542849745720732, 8.102622301357312758748216967144, 9.125792423688083910032050796479, 9.291254042657563232541240983018