| L(s) = 1 | − 4.55·2-s − 5.72·3-s + 12.7·4-s + 17.5·5-s + 26.0·6-s − 7·7-s − 21.5·8-s + 5.72·9-s − 80.0·10-s + 26.0·11-s − 72.8·12-s − 42.9·13-s + 31.8·14-s − 100.·15-s − 3.85·16-s − 56.6·17-s − 26.0·18-s + 102.·19-s + 223.·20-s + 40.0·21-s − 118.·22-s − 23·23-s + 123.·24-s + 184.·25-s + 195.·26-s + 121.·27-s − 89.0·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 1.10·3-s + 1.59·4-s + 1.57·5-s + 1.77·6-s − 0.377·7-s − 0.950·8-s + 0.212·9-s − 2.53·10-s + 0.713·11-s − 1.75·12-s − 0.916·13-s + 0.608·14-s − 1.73·15-s − 0.0602·16-s − 0.807·17-s − 0.341·18-s + 1.24·19-s + 2.50·20-s + 0.416·21-s − 1.14·22-s − 0.208·23-s + 1.04·24-s + 1.47·25-s + 1.47·26-s + 0.867·27-s − 0.601·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6077602063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6077602063\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 4.55T + 8T^{2} \) |
| 3 | \( 1 + 5.72T + 27T^{2} \) |
| 5 | \( 1 - 17.5T + 125T^{2} \) |
| 11 | \( 1 - 26.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 59.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 249.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 263.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 210.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 446.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 884.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 415.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 506.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 837.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 565.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 20.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 701.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.85T + 7.04e5T^{2} \) |
| 97 | \( 1 + 165.T + 9.12e5T^{2} \) |
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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
−11.94831884402593704714748323614, −11.16600820979760910630464435952, −9.908925751016919868040863615973, −9.778968077145821689447111548180, −8.601216237301081968724163473873, −6.98313760084618400765662379684, −6.29451506140610242456471126558, −5.14513655524434214994296502823, −2.31632463470666705116842478073, −0.843085375051097346602180564176,
0.843085375051097346602180564176, 2.31632463470666705116842478073, 5.14513655524434214994296502823, 6.29451506140610242456471126558, 6.98313760084618400765662379684, 8.601216237301081968724163473873, 9.778968077145821689447111548180, 9.908925751016919868040863615973, 11.16600820979760910630464435952, 11.94831884402593704714748323614
