| L(s) = 1 | − 27i·3-s − 349. i·5-s + 856.·7-s − 729·9-s − 7.46e3i·11-s − 7.08e3i·13-s − 9.44e3·15-s − 3.05e4·17-s − 1.62e4i·19-s − 2.31e4i·21-s − 6.64e3·23-s − 4.43e4·25-s + 1.96e4i·27-s + 2.16e5i·29-s − 1.61e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.25i·5-s + 0.943·7-s − 0.333·9-s − 1.69i·11-s − 0.894i·13-s − 0.722·15-s − 1.50·17-s − 0.543i·19-s − 0.544i·21-s − 0.113·23-s − 0.567·25-s + 0.192i·27-s + 1.64i·29-s − 0.971·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.464963211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464963211\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| good | 5 | \( 1 + 349. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 856.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.46e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 7.08e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.05e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.62e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 6.64e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.16e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.61e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.02e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 4.53e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.45e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.22e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.94e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.81e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.79e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 5.11e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.14e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.74e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 6.73e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.60e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.165667040837562482058970727655, −8.570260876939410408959607125588, −8.015458674124174050482048522600, −6.75146430913312510437414917487, −5.51890621257926709534415590708, −4.95943978520158437148197036114, −3.58424561752324096155816378156, −2.15750440868663689459330443578, −1.01109661203314993433212472091, −0.32108612971596459784882487334,
1.82133618961408848401421265961, 2.53500342971031941010919880021, 4.13707548078428851141252998448, 4.60531734624067992015622144444, 6.09154392460582762399682887841, 7.03820612201369041207741871204, 7.79538282338937366935199634878, 9.091275050026756773910992400845, 9.880699068308966166682960678915, 10.78844002875767977936279829332
