| L(s) = 1 | − 46.7·3-s − 1.20e3i·5-s − 1.13e3i·7-s + 2.18e3·9-s − 2.06e4·11-s − 2.57e4i·13-s + 5.63e4i·15-s + 2.47e3·17-s + 2.06e5·19-s + 5.30e4i·21-s + 2.23e5i·23-s − 1.06e6·25-s − 1.02e5·27-s + 2.02e5i·29-s + 1.16e6i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.92i·5-s − 0.472i·7-s + 0.333·9-s − 1.40·11-s − 0.901i·13-s + 1.11i·15-s + 0.0296·17-s + 1.58·19-s + 0.272i·21-s + 0.797i·23-s − 2.71·25-s − 0.192·27-s + 0.286i·29-s + 1.26i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.049113330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049113330\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7T \) |
| good | 5 | \( 1 + 1.20e3iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.13e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.06e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 2.57e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 2.47e3T + 6.97e9T^{2} \) |
| 19 | \( 1 - 2.06e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.23e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 2.02e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.16e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.62e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 1.46e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.00e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 6.93e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.16e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 2.03e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + 3.06e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 3.74e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.03e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.81e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.18e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 8.39e6T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.73e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.34e8T + 7.83e15T^{2} \) |
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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
−9.953186777483457129923808386852, −9.124949945205442989524548237508, −7.999180883298495535386269765728, −7.50248496475327760531895370823, −5.74231008923535194436119255819, −5.22699056284677208011368563214, −4.50083472106077689581675339366, −3.08725411369295373274197492763, −1.34448366616894379798602868686, −0.76699947850529358089875820153,
0.30329793256587393163377211997, 2.18184147387089328856920231869, 2.81457050245772211507745701802, 4.03685934695977647921708761621, 5.47468749437512449715015721410, 6.16507132827724186617523176041, 7.23191354573313813315510795863, 7.71567535050016561094148788380, 9.329703017705386589618332071432, 10.20260117544264687583975998552
