| L(s) = 1 | + (24.3 + 77.2i)3-s + (−547. − 301. i)5-s + 1.53e3i·7-s + (−5.37e3 + 3.75e3i)9-s + 7.76e3i·11-s + 2.43e4i·13-s + (1.00e4 − 4.96e4i)15-s + 8.39e4·17-s − 1.15e5·19-s + (−1.18e5 + 3.73e4i)21-s − 1.86e5·23-s + (2.08e5 + 3.30e5i)25-s + (−4.21e5 − 3.24e5i)27-s − 9.07e5i·29-s − 1.08e6·31-s + ⋯ |
| L(s) = 1 | + (0.300 + 0.953i)3-s + (−0.875 − 0.483i)5-s + 0.639i·7-s + (−0.819 + 0.572i)9-s + 0.530i·11-s + 0.852i·13-s + (0.197 − 0.980i)15-s + 1.00·17-s − 0.884·19-s + (−0.610 + 0.192i)21-s − 0.665·23-s + (0.533 + 0.845i)25-s + (−0.792 − 0.610i)27-s − 1.28i·29-s − 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.08128583377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08128583377\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-24.3 - 77.2i)T \) |
| 5 | \( 1 + (547. + 301. i)T \) |
| good | 7 | \( 1 - 1.53e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 7.76e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.43e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 8.39e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.15e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.86e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 9.07e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.08e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 5.92e4iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 3.29e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.48e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 4.73e5T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.18e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 3.27e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.05e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.13e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 9.40e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.64e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.17e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.83e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.11e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 5.09e7iT - 7.83e15T^{2} \) |
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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
−11.70336560295853236716372787346, −10.51321680260318416015672956332, −9.370057342747562127881053607619, −8.586078301253450928924861682358, −7.51713983869263115264760590188, −5.76300434736602857948355513464, −4.55001085678816099873914865244, −3.68073561996253154707866443314, −2.10947347108093425459138519738, −0.02270094810783094656616286281,
1.13667131796250798407125119858, 2.86180254617075050955892672844, 3.85050672142763312929488316786, 5.74001052452216012723094637523, 7.00712880583001600167498665162, 7.77974439001323223318739013010, 8.642774724776597581353467007510, 10.32516650500872356917690304845, 11.23683703318445964762891512521, 12.31030646424778939531241166127
