| L(s) = 1 | + (1.57 − 2.55i)3-s + 1.31i·5-s + 10.2·7-s + (−4.01 − 8.05i)9-s + 16.6i·11-s + 18.7·13-s + (3.35 + 2.07i)15-s − 4.38i·17-s − 11.5·19-s + (16.1 − 26.1i)21-s − 16.7i·23-s + 23.2·25-s + (−26.8 − 2.46i)27-s + 12.5i·29-s + 20.3·31-s + ⋯ |
| L(s) = 1 | + (0.526 − 0.850i)3-s + 0.263i·5-s + 1.46·7-s + (−0.446 − 0.894i)9-s + 1.51i·11-s + 1.44·13-s + (0.223 + 0.138i)15-s − 0.257i·17-s − 0.608·19-s + (0.769 − 1.24i)21-s − 0.728i·23-s + 0.930·25-s + (−0.995 − 0.0912i)27-s + 0.432i·29-s + 0.655·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.32007 - 0.659696i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32007 - 0.659696i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.57 + 2.55i)T \) |
| good | 5 | \( 1 - 1.31iT - 25T^{2} \) |
| 7 | \( 1 - 10.2T + 49T^{2} \) |
| 11 | \( 1 - 16.6iT - 121T^{2} \) |
| 13 | \( 1 - 18.7T + 169T^{2} \) |
| 17 | \( 1 + 4.38iT - 289T^{2} \) |
| 19 | \( 1 + 11.5T + 361T^{2} \) |
| 23 | \( 1 + 16.7iT - 529T^{2} \) |
| 29 | \( 1 - 12.5iT - 841T^{2} \) |
| 31 | \( 1 - 20.3T + 961T^{2} \) |
| 37 | \( 1 + 18.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 78.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 81.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 29.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 72.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 56.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 136. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 80.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 86.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 80.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 20.4T + 9.40e3T^{2} \) |
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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
−11.13451320165623693322802410592, −10.24252196814431691725783216799, −8.808199227736071822289731291909, −8.337174547201754164131395558996, −7.27867499574315947249242483276, −6.56828240645447455160634339567, −5.14116624983839520589520051126, −3.96985038582627626565472470330, −2.35579461991162504429855341847, −1.35698355360052007025516023476,
1.40788260227451033975989848682, 3.12681041232824466861589936992, 4.20880378904729459461227640765, 5.19873169045657921476417658878, 6.20376661905466437577607806159, 8.051260817303166523899532482966, 8.375556887599032991756926149450, 9.127134435079989266223498881241, 10.53724561258202835708454567425, 11.05452865189262873620272455445
