| L(s) = 1 | + 516.·2-s + 1.59e4·3-s + 1.35e5·4-s + 1.06e6·5-s + 8.25e6·6-s + 2.48e7·7-s + 2.13e6·8-s + 1.26e8·9-s + 5.51e8·10-s − 1.01e9·11-s + 2.16e9·12-s − 1.98e9·13-s + 1.28e10·14-s + 1.71e10·15-s − 1.66e10·16-s + 6.97e9·17-s + 6.53e10·18-s − 1.19e11·19-s + 1.44e11·20-s + 3.97e11·21-s − 5.22e11·22-s + 5.69e11·23-s + 3.42e10·24-s + 3.80e11·25-s − 1.02e12·26-s − 4.14e10·27-s + 3.36e12·28-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.40·3-s + 1.03·4-s + 1.22·5-s + 2.00·6-s + 1.63·7-s + 0.0450·8-s + 0.979·9-s + 1.74·10-s − 1.42·11-s + 1.45·12-s − 0.676·13-s + 2.32·14-s + 1.72·15-s − 0.967·16-s + 0.242·17-s + 1.39·18-s − 1.61·19-s + 1.26·20-s + 2.29·21-s − 2.03·22-s + 1.51·23-s + 0.0634·24-s + 0.499·25-s − 0.963·26-s − 0.0282·27-s + 1.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(8.251937569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.251937569\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - 6.97e9T \) |
| good | 2 | \( 1 - 516.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.59e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.06e6T + 7.62e11T^{2} \) |
| 7 | \( 1 - 2.48e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.01e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.98e9T + 8.65e18T^{2} \) |
| 19 | \( 1 + 1.19e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.69e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.32e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 3.32e11T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.58e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 2.80e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 6.82e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 7.06e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.91e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 8.60e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.78e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 3.38e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 3.42e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 7.14e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.01e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 9.29e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 6.47e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 2.22e16T + 5.95e33T^{2} \) |
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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
−14.68138117989006345619044312712, −13.59934909419896447803904226520, −12.86789233798930749275178378547, −10.84814912910858691209901424620, −9.073124136176089452822376626496, −7.68833421109880665877877477040, −5.57555967086825114112862237074, −4.50994661105748138201255680259, −2.65803526945908971430445730046, −2.02991435842645054917862469667,
2.02991435842645054917862469667, 2.65803526945908971430445730046, 4.50994661105748138201255680259, 5.57555967086825114112862237074, 7.68833421109880665877877477040, 9.073124136176089452822376626496, 10.84814912910858691209901424620, 12.86789233798930749275178378547, 13.59934909419896447803904226520, 14.68138117989006345619044312712
