| L(s) = 1 | − 3.26·2-s + 66.7·3-s − 117.·4-s + 259.·5-s − 217.·6-s + 1.45e3·7-s + 800.·8-s + 2.26e3·9-s − 847.·10-s − 4.45e3·11-s − 7.83e3·12-s − 2.19e3·13-s − 4.74e3·14-s + 1.73e4·15-s + 1.24e4·16-s − 1.92e4·17-s − 7.39e3·18-s − 3.93e4·19-s − 3.05e4·20-s + 9.70e4·21-s + 1.45e4·22-s + 2.84e4·23-s + 5.34e4·24-s − 1.05e4·25-s + 7.16e3·26-s + 5.48e3·27-s − 1.70e5·28-s + ⋯ |
| L(s) = 1 | − 0.288·2-s + 1.42·3-s − 0.916·4-s + 0.930·5-s − 0.411·6-s + 1.60·7-s + 0.552·8-s + 1.03·9-s − 0.268·10-s − 1.00·11-s − 1.30·12-s − 0.277·13-s − 0.461·14-s + 1.32·15-s + 0.757·16-s − 0.950·17-s − 0.299·18-s − 1.31·19-s − 0.852·20-s + 2.28·21-s + 0.290·22-s + 0.487·23-s + 0.788·24-s − 0.134·25-s + 0.0799·26-s + 0.0536·27-s − 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.912500128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912500128\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 + 3.26T + 128T^{2} \) |
| 3 | \( 1 - 66.7T + 2.18e3T^{2} \) |
| 5 | \( 1 - 259.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.45e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.45e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.93e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.84e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.46e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.00e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.90e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.59e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.77e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.63e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.18e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.69e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.77e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 8.77e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.74e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.82e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.51e6T + 8.07e13T^{2} \) |
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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
−18.17625549828396254413366699335, −17.42925819683807755491738236296, −15.00826596313836769647315448184, −14.06156804850229951113202475944, −13.18682776790803713357281517260, −10.44086974241802151174872736820, −8.897448735203107419215443771410, −8.035869898238293724267354702106, −4.76558367138800486027837590604, −2.08403255383102771431664916312,
2.08403255383102771431664916312, 4.76558367138800486027837590604, 8.035869898238293724267354702106, 8.897448735203107419215443771410, 10.44086974241802151174872736820, 13.18682776790803713357281517260, 14.06156804850229951113202475944, 15.00826596313836769647315448184, 17.42925819683807755491738236296, 18.17625549828396254413366699335
