| L(s) = 1 | + 2·2-s − 28·4-s − 11·5-s − 120·8-s − 22·10-s − 269·11-s − 308·13-s + 656·16-s − 1.89e3·17-s − 164·19-s + 308·20-s − 538·22-s + 3.26e3·23-s − 3.00e3·25-s − 616·26-s − 2.41e3·29-s + 2.84e3·31-s + 5.15e3·32-s − 3.79e3·34-s − 1.13e4·37-s − 328·38-s + 1.32e3·40-s + 1.68e4·41-s − 7.89e3·43-s + 7.53e3·44-s + 6.52e3·46-s − 2.11e4·47-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s − 0.196·5-s − 0.662·8-s − 0.0695·10-s − 0.670·11-s − 0.505·13-s + 0.640·16-s − 1.59·17-s − 0.104·19-s + 0.172·20-s − 0.236·22-s + 1.28·23-s − 0.961·25-s − 0.178·26-s − 0.533·29-s + 0.530·31-s + 0.889·32-s − 0.562·34-s − 1.36·37-s − 0.0368·38-s + 0.130·40-s + 1.56·41-s − 0.651·43-s + 0.586·44-s + 0.454·46-s − 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.025185959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025185959\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 5 | \( 1 + 11 T + p^{5} T^{2} \) |
| 11 | \( 1 + 269 T + p^{5} T^{2} \) |
| 13 | \( 1 + 308 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1896 T + p^{5} T^{2} \) |
| 19 | \( 1 + 164 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3264 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2417 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2841 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11328 T + p^{5} T^{2} \) |
| 41 | \( 1 - 16856 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7894 T + p^{5} T^{2} \) |
| 47 | \( 1 + 21102 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29691 T + p^{5} T^{2} \) |
| 59 | \( 1 - 8163 T + p^{5} T^{2} \) |
| 61 | \( 1 - 15166 T + p^{5} T^{2} \) |
| 67 | \( 1 + 32078 T + p^{5} T^{2} \) |
| 71 | \( 1 - 38274 T + p^{5} T^{2} \) |
| 73 | \( 1 - 34866 T + p^{5} T^{2} \) |
| 79 | \( 1 - 13529 T + p^{5} T^{2} \) |
| 83 | \( 1 - 68103 T + p^{5} T^{2} \) |
| 89 | \( 1 - 114922 T + p^{5} T^{2} \) |
| 97 | \( 1 - 154959 T + p^{5} T^{2} \) |
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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
−10.29273593753359442714264281834, −9.312523009832640561866485308561, −8.611192527019822928221568068930, −7.61514901050822142364310534053, −6.51056653555152851260743999809, −5.29621828875002232903808259837, −4.59900992130599715148023906059, −3.53379589401821721882330478726, −2.27315949575493103006232454174, −0.48036742661261219661656768673,
0.48036742661261219661656768673, 2.27315949575493103006232454174, 3.53379589401821721882330478726, 4.59900992130599715148023906059, 5.29621828875002232903808259837, 6.51056653555152851260743999809, 7.61514901050822142364310534053, 8.611192527019822928221568068930, 9.312523009832640561866485308561, 10.29273593753359442714264281834
