L(s) = 1 | − 16.8·2-s − 116.·3-s − 229.·4-s − 1.10e3·5-s + 1.96e3·6-s − 5.16e3·7-s + 1.24e4·8-s − 6.02e3·9-s + 1.85e4·10-s + 4.45e4·11-s + 2.68e4·12-s + 6.96e4·13-s + 8.68e4·14-s + 1.28e5·15-s − 9.20e4·16-s + 8.35e4·17-s + 1.01e5·18-s − 1.70e5·19-s + 2.53e5·20-s + 6.03e5·21-s − 7.48e5·22-s − 2.00e6·23-s − 1.45e6·24-s − 7.35e5·25-s − 1.17e6·26-s + 3.00e6·27-s + 1.18e6·28-s + ⋯ |
L(s) = 1 | − 0.742·2-s − 0.833·3-s − 0.447·4-s − 0.789·5-s + 0.619·6-s − 0.812·7-s + 1.07·8-s − 0.305·9-s + 0.586·10-s + 0.917·11-s + 0.373·12-s + 0.676·13-s + 0.604·14-s + 0.657·15-s − 0.351·16-s + 0.242·17-s + 0.227·18-s − 0.299·19-s + 0.353·20-s + 0.677·21-s − 0.681·22-s − 1.49·23-s − 0.896·24-s − 0.376·25-s − 0.502·26-s + 1.08·27-s + 0.364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.4230817474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4230817474\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - 8.35e4T \) |
good | 2 | \( 1 + 16.8T + 512T^{2} \) |
| 3 | \( 1 + 116.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.10e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.16e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.96e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.55e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.51e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.59e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.49e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.36e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.50e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.94e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.27e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.88e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.32e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.38e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.74e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.82e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.03e9T + 7.60e17T^{2} \) |
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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
−16.85937991517856665729287539245, −15.93788502167115774254983376808, −14.01540939448793619274921657167, −12.34076933170699821326559890104, −11.07527062996211902781796250507, −9.569886912137996803937703206619, −8.099615789853963887761836891481, −6.19883438744100674546912363288, −4.05334358887555833762083259030, −0.62618691271467735594292864210,
0.62618691271467735594292864210, 4.05334358887555833762083259030, 6.19883438744100674546912363288, 8.099615789853963887761836891481, 9.569886912137996803937703206619, 11.07527062996211902781796250507, 12.34076933170699821326559890104, 14.01540939448793619274921657167, 15.93788502167115774254983376808, 16.85937991517856665729287539245