| L(s) = 1 | − 0.484·2-s − 652.·3-s − 2.04e3·4-s − 5.43e3·5-s + 316.·6-s − 5.96e4·7-s + 1.98e3·8-s + 2.49e5·9-s + 2.63e3·10-s + 1.06e5·11-s + 1.33e6·12-s − 1.39e6·13-s + 2.88e4·14-s + 3.54e6·15-s + 4.19e6·16-s − 1.41e6·17-s − 1.20e5·18-s − 1.25e7·19-s + 1.11e7·20-s + 3.89e7·21-s − 5.16e4·22-s + 3.76e7·23-s − 1.29e6·24-s − 1.92e7·25-s + 6.73e5·26-s − 4.70e7·27-s + 1.22e8·28-s + ⋯ |
| L(s) = 1 | − 0.0107·2-s − 1.55·3-s − 0.999·4-s − 0.777·5-s + 0.0166·6-s − 1.34·7-s + 0.0214·8-s + 1.40·9-s + 0.00832·10-s + 0.199·11-s + 1.55·12-s − 1.03·13-s + 0.0143·14-s + 1.20·15-s + 0.999·16-s − 0.242·17-s − 0.0150·18-s − 1.16·19-s + 0.777·20-s + 2.08·21-s − 0.00213·22-s + 1.21·23-s − 0.0332·24-s − 0.394·25-s + 0.0111·26-s − 0.631·27-s + 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.1118394927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1118394927\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + 1.41e6T \) |
| good | 2 | \( 1 + 0.484T + 2.04e3T^{2} \) |
| 3 | \( 1 + 652.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 5.43e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 5.96e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.06e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.39e6T + 1.79e12T^{2} \) |
| 19 | \( 1 + 1.25e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.76e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.71e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.21e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.34e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.81e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 9.76e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.00e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.49e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 2.83e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 6.00e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 4.62e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.22e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 8.17e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.35e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.90e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.47e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.64e11T + 7.15e21T^{2} \) |
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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
−16.59326801278720130047850411142, −15.11987558208591995320788493356, −13.01608545273353016283963453903, −12.22890521176162144596425219983, −10.70102925852808499882830265739, −9.286573210885153366168057903560, −7.04635092134536454437507481376, −5.46745872881936824352808693909, −3.97069246614107128276292164698, −0.26239216023185102926149833675,
0.26239216023185102926149833675, 3.97069246614107128276292164698, 5.46745872881936824352808693909, 7.04635092134536454437507481376, 9.286573210885153366168057903560, 10.70102925852808499882830265739, 12.22890521176162144596425219983, 13.01608545273353016283963453903, 15.11987558208591995320788493356, 16.59326801278720130047850411142
