| L(s) = 1 | + 11.1·2-s + 81·3-s − 386.·4-s − 1.76e3·5-s + 906.·6-s − 1.00e4·8-s + 6.56e3·9-s − 1.97e4·10-s − 3.82e4·11-s − 3.13e4·12-s − 4.93e4·13-s − 1.43e5·15-s + 8.55e4·16-s + 1.97e5·17-s + 7.33e4·18-s − 8.95e5·19-s + 6.84e5·20-s − 4.27e5·22-s − 2.91e5·23-s − 8.14e5·24-s + 1.17e6·25-s − 5.51e5·26-s + 5.31e5·27-s − 5.89e6·29-s − 1.60e6·30-s − 3.29e5·31-s + 6.10e6·32-s + ⋯ |
| L(s) = 1 | + 0.494·2-s + 0.577·3-s − 0.755·4-s − 1.26·5-s + 0.285·6-s − 0.867·8-s + 0.333·9-s − 0.625·10-s − 0.786·11-s − 0.436·12-s − 0.479·13-s − 0.730·15-s + 0.326·16-s + 0.572·17-s + 0.164·18-s − 1.57·19-s + 0.956·20-s − 0.388·22-s − 0.217·23-s − 0.501·24-s + 0.601·25-s − 0.236·26-s + 0.192·27-s − 1.54·29-s − 0.361·30-s − 0.0641·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.111896035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.111896035\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 11.1T + 512T^{2} \) |
| 5 | \( 1 + 1.76e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 3.82e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.97e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.95e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.91e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.89e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.29e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.18e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.34e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.61e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.00e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.73e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.33e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.39e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.81e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.53e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.98e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.85e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.55e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.01e9T + 7.60e17T^{2} \) |
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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
−11.52284299801562120358158063728, −10.29453085065361657386471931335, −9.105547460487036483283632025318, −8.157786700585966679900212509718, −7.41977086547084928273570243183, −5.71618669023529304239438049737, −4.41086367873558418203844057153, −3.77051262776606581413695085437, −2.50687051285424976421375385563, −0.47011152740167823238638105364,
0.47011152740167823238638105364, 2.50687051285424976421375385563, 3.77051262776606581413695085437, 4.41086367873558418203844057153, 5.71618669023529304239438049737, 7.41977086547084928273570243183, 8.157786700585966679900212509718, 9.105547460487036483283632025318, 10.29453085065361657386471931335, 11.52284299801562120358158063728
