| L(s) = 1 | − 46.7i·3-s − 966.·5-s − 1.23e3i·7-s − 2.18e3·9-s + 6.45e3i·11-s + 2.53e4·13-s + 4.52e4i·15-s − 4.39e4·17-s − 7.92e4i·19-s − 5.77e4·21-s + 1.70e5i·23-s + 5.43e5·25-s + 1.02e5i·27-s − 3.47e5·29-s − 1.63e6i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.54·5-s − 0.514i·7-s − 0.333·9-s + 0.440i·11-s + 0.886·13-s + 0.892i·15-s − 0.526·17-s − 0.608i·19-s − 0.297·21-s + 0.608i·23-s + 1.39·25-s + 0.192i·27-s − 0.490·29-s − 1.76i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6977599956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6977599956\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 + 966.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 1.23e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 6.45e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.53e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 4.39e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 7.92e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.70e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.47e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.63e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.64e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.82e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.10e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.17e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.03e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.46e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 5.57e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.50e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.43e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.72e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.66e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 2.20e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 5.90e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.52e8T + 7.83e15T^{2} \) |
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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
−9.368454082384897276477115288063, −8.375666058049661394813832443211, −7.57192926410367366944064472661, −7.02569070902046313934045017213, −5.81663330192009823504601662186, −4.33634642990140888208682885765, −3.78321817860892351367393799406, −2.44297063293035336961069959121, −0.986541929714702473694879037244, −0.18834277737326014138915435022,
1.00684376105254884306504764418, 2.74236967445529980927426830422, 3.72735484177949068487559952581, 4.40921016663837841041627694106, 5.61908530439974245709651599277, 6.70254164119423704530972606942, 7.946546921775101554732968932343, 8.504729644313347541465869063101, 9.368806147122678926279774700390, 10.77367491626623839154730087896
