| L(s) = 1 | + 49.8·2-s − 593.·3-s + 441.·4-s + 6.65e3·5-s − 2.96e4·6-s + 3.17e4·7-s − 8.01e4·8-s + 1.75e5·9-s + 3.32e5·10-s + 1.02e6·11-s − 2.62e5·12-s + 1.73e6·13-s + 1.58e6·14-s − 3.95e6·15-s − 4.90e6·16-s − 1.41e6·17-s + 8.76e6·18-s + 1.31e7·19-s + 2.93e6·20-s − 1.88e7·21-s + 5.09e7·22-s − 2.36e7·23-s + 4.76e7·24-s − 4.51e6·25-s + 8.66e7·26-s + 8.69e5·27-s + 1.40e7·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 1.41·3-s + 0.215·4-s + 0.952·5-s − 1.55·6-s + 0.713·7-s − 0.864·8-s + 0.991·9-s + 1.05·10-s + 1.91·11-s − 0.304·12-s + 1.29·13-s + 0.786·14-s − 1.34·15-s − 1.16·16-s − 0.242·17-s + 1.09·18-s + 1.21·19-s + 0.205·20-s − 1.00·21-s + 2.10·22-s − 0.766·23-s + 1.22·24-s − 0.0925·25-s + 1.42·26-s + 0.0116·27-s + 0.153·28-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
| \(L(6)\) |
\(\approx\) |
\(2.457591922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.457591922\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 49.8T + 2.04e3T^{2} \) |
| 3 | \( 1 + 593.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 6.65e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.17e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.02e6T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.73e6T + 1.79e12T^{2} \) |
| 19 | \( 1 - 1.31e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.36e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 2.15e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.80e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.82e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 3.71e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.70e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.88e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 6.64e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.24e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 2.79e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.60e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.17e8T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.00e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.09e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.24e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.44e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.30e11T + 7.15e21T^{2} \) |
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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
−16.28786838654514061276740659757, −14.49615807547610313010218166303, −13.56303362669384851842683437326, −12.03852757124295273677543478533, −11.22174554124523403267695713992, −9.281499835910213635918853555338, −6.33653476860420911061959308731, −5.60335604652473392654728607288, −4.10523952491093379779778233222, −1.25075706498458769086125781744,
1.25075706498458769086125781744, 4.10523952491093379779778233222, 5.60335604652473392654728607288, 6.33653476860420911061959308731, 9.281499835910213635918853555338, 11.22174554124523403267695713992, 12.03852757124295273677543478533, 13.56303362669384851842683437326, 14.49615807547610313010218166303, 16.28786838654514061276740659757
