| L(s) = 1 | + 115. i·3-s − 1.17e3·5-s − 4.24e3i·7-s − 6.72e3·9-s − 4.23e3i·11-s + 2.19e4·13-s − 1.35e5i·15-s − 1.14e5·17-s + 4.42e4i·19-s + 4.89e5·21-s + 1.14e5i·23-s + 9.93e5·25-s − 1.88e4i·27-s + 2.62e5·29-s − 1.08e6i·31-s + ⋯ |
| L(s) = 1 | + 1.42i·3-s − 1.88·5-s − 1.76i·7-s − 1.02·9-s − 0.289i·11-s + 0.767·13-s − 2.67i·15-s − 1.36·17-s + 0.339i·19-s + 2.51·21-s + 0.408i·23-s + 2.54·25-s − 0.0354i·27-s + 0.371·29-s − 1.17i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8363610878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8363610878\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 115. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 1.17e3T + 3.90e5T^{2} \) |
| 7 | \( 1 + 4.24e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 4.23e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.19e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.14e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 4.42e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.14e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 2.62e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.08e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.55e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 4.94e5T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.26e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 1.22e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 4.49e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 3.25e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.91e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.12e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.09e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.41e5T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.21e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 6.37e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 9.29e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 9.60e7T + 7.83e15T^{2} \) |
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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
−11.53167019120129929159144606486, −10.96799230384640835651755037530, −10.27631738804935539044926133695, −8.857145208613247891557412064624, −7.86753755047855399210435755871, −6.82326417416801212865949637376, −4.75314528628727682884480686113, −3.93036946898340502989589033590, −3.57610255491418917844461038438, −0.68825443992436687208084578509,
0.36461743766912639929932341169, 1.91908176706597049453048053949, 3.15954718675775442956346314448, 4.77555502676182475518497315193, 6.36709474451051317638920881556, 7.19361613578067076303133476984, 8.443277818885448969780677267593, 8.699934187677268273582243049638, 11.03057654536306077404217688159, 11.86711630991973484155829088416
