| L(s) = 1 | + 73.5·2-s + 3.35e3·4-s + 3.41e3·5-s − 5.41e4·7-s + 9.61e4·8-s + 2.51e5·10-s − 6.46e5·11-s − 2.07e4·13-s − 3.98e6·14-s + 1.93e5·16-s + 5.43e6·17-s − 1.12e7·19-s + 1.14e7·20-s − 4.75e7·22-s − 5.06e7·23-s − 3.71e7·25-s − 1.52e6·26-s − 1.81e8·28-s − 5.57e7·29-s + 2.69e8·31-s − 1.82e8·32-s + 3.99e8·34-s − 1.85e8·35-s − 2.61e8·37-s − 8.25e8·38-s + 3.28e8·40-s + 9.21e8·41-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.63·4-s + 0.489·5-s − 1.21·7-s + 1.03·8-s + 0.794·10-s − 1.21·11-s − 0.0154·13-s − 1.97·14-s + 0.0461·16-s + 0.928·17-s − 1.04·19-s + 0.801·20-s − 1.96·22-s − 1.64·23-s − 0.760·25-s − 0.0251·26-s − 1.99·28-s − 0.504·29-s + 1.68·31-s − 0.962·32-s + 1.50·34-s − 0.596·35-s − 0.620·37-s − 1.69·38-s + 0.507·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 73.5T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.41e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 5.41e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.46e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.07e4T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.43e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.12e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.06e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 5.57e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.69e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.61e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.21e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.86e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.85e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.53e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 5.61e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 3.69e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 2.50e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 5.97e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.54e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.94e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.71e8T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.91e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 3.54e10T + 7.15e21T^{2} \) |
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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
−12.10482496722081550283859314971, −10.60512044335961466595874586791, −9.645407913524589772179263878584, −7.80324946067355830089997042108, −6.28338095459945264795764667003, −5.75484397839954307474997365696, −4.36364934106021964997147443628, −3.16661197482429378571005642602, −2.20943009661125346001579692954, 0,
2.20943009661125346001579692954, 3.16661197482429378571005642602, 4.36364934106021964997147443628, 5.75484397839954307474997365696, 6.28338095459945264795764667003, 7.80324946067355830089997042108, 9.645407913524589772179263878584, 10.60512044335961466595874586791, 12.10482496722081550283859314971
