| L(s) = 1 | − 46.7i·3-s + 1.08e3·5-s − 399. i·7-s − 2.18e3·9-s + 4.27e3i·11-s + 3.80e4·13-s − 5.06e4i·15-s − 1.25e5·17-s + 2.16e5i·19-s − 1.86e4·21-s − 3.00e5i·23-s + 7.83e5·25-s + 1.02e5i·27-s + 4.34e5·29-s + 7.87e5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.73·5-s − 0.166i·7-s − 0.333·9-s + 0.291i·11-s + 1.33·13-s − 1.00i·15-s − 1.49·17-s + 1.66i·19-s − 0.0959·21-s − 1.07i·23-s + 2.00·25-s + 0.192i·27-s + 0.613·29-s + 0.852i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.531674954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.531674954\) |
| \(L(5)\) |
|
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 | \( 1 - 1.08e3T + 3.90e5T^{2} \) |
| 7 | \( 1 + 399. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 4.27e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.80e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.25e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 2.16e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 3.00e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 4.34e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 7.87e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.35e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 6.23e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.76e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 7.12e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 7.12e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.33e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.19e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.18e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.35e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.79e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.52e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 2.93e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 9.53e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 7.13e7T + 7.83e15T^{2} \) |
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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
−10.13410810490659508166155734088, −8.946432986425875097681005903715, −8.364842832741197784023226235526, −6.84350458081665681089337942924, −6.26729558217675003194165233622, −5.50117378476898111337695486983, −4.16404767863965863698009373656, −2.65431456252932592942453527109, −1.79475576116129721145749966592, −0.999777322460346457721797000163,
0.73486218545875546866548979840, 1.98114825613148240868451702564, 2.84166395858256002204729547262, 4.22648223272530782496542400492, 5.33422584358392398580805976193, 6.06877808058984793448938026893, 6.87512279708842604103230785688, 8.605018149940217005173601261969, 9.113856895677615445264066523913, 9.862361402742023544149359450847
