| L(s) = 1 | − 5.53e3·2-s − 4.78e6·3-s − 5.06e8·4-s − 9.04e9·5-s + 2.64e10·6-s + 6.78e11·7-s + 5.77e12·8-s + 2.28e13·9-s + 5.00e13·10-s − 1.24e15·11-s + 2.42e15·12-s − 2.11e15·13-s − 3.75e15·14-s + 4.32e16·15-s + 2.39e17·16-s + 2.87e17·17-s − 1.26e17·18-s − 1.38e18·19-s + 4.58e18·20-s − 3.24e18·21-s + 6.91e18·22-s − 6.33e19·23-s − 2.76e19·24-s − 1.04e20·25-s + 1.17e19·26-s − 1.09e20·27-s − 3.43e20·28-s + ⋯ |
| L(s) = 1 | − 0.238·2-s − 0.577·3-s − 0.942·4-s − 0.662·5-s + 0.137·6-s + 0.377·7-s + 0.464·8-s + 0.333·9-s + 0.158·10-s − 0.991·11-s + 0.544·12-s − 0.148·13-s − 0.0903·14-s + 0.382·15-s + 0.832·16-s + 0.413·17-s − 0.0796·18-s − 0.399·19-s + 0.625·20-s − 0.218·21-s + 0.236·22-s − 1.13·23-s − 0.268·24-s − 0.560·25-s + 0.0355·26-s − 0.192·27-s − 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.1922188035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1922188035\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4.78e6T \) |
| 7 | \( 1 - 6.78e11T \) |
| good | 2 | \( 1 + 5.53e3T + 5.36e8T^{2} \) |
| 5 | \( 1 + 9.04e9T + 1.86e20T^{2} \) |
| 11 | \( 1 + 1.24e15T + 1.58e30T^{2} \) |
| 13 | \( 1 + 2.11e15T + 2.01e32T^{2} \) |
| 17 | \( 1 - 2.87e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 1.38e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 6.33e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 8.68e19T + 2.56e42T^{2} \) |
| 31 | \( 1 + 8.17e21T + 1.77e43T^{2} \) |
| 37 | \( 1 + 4.04e21T + 3.00e45T^{2} \) |
| 41 | \( 1 + 1.80e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 3.59e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + 2.81e23T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.29e25T + 1.00e50T^{2} \) |
| 59 | \( 1 - 1.72e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 7.74e25T + 5.95e51T^{2} \) |
| 67 | \( 1 - 9.27e25T + 9.04e52T^{2} \) |
| 71 | \( 1 - 4.90e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + 1.35e27T + 1.08e54T^{2} \) |
| 79 | \( 1 - 9.34e26T + 1.07e55T^{2} \) |
| 83 | \( 1 + 2.31e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + 2.85e28T + 3.40e56T^{2} \) |
| 97 | \( 1 - 9.87e28T + 4.13e57T^{2} \) |
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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
−12.14929072798743280046804276717, −10.85956788916189412531518202450, −9.796288590013697831441732522021, −8.330575816629642303005785821405, −7.49797575241293062834265439961, −5.67671909652084669432978461202, −4.69055018126854644800165922638, −3.60027104604450395996097407959, −1.72202696557222229566760157688, −0.22206477937588874911561916746,
0.22206477937588874911561916746, 1.72202696557222229566760157688, 3.60027104604450395996097407959, 4.69055018126854644800165922638, 5.67671909652084669432978461202, 7.49797575241293062834265439961, 8.330575816629642303005785821405, 9.796288590013697831441732522021, 10.85956788916189412531518202450, 12.14929072798743280046804276717
