| L(s) = 1 | + 9.23·2-s − 23.1·3-s + 53.2·4-s − 213.·6-s + 161.·7-s + 196.·8-s + 290.·9-s − 66.6·11-s − 1.23e3·12-s + 169·13-s + 1.48e3·14-s + 107.·16-s − 536.·17-s + 2.68e3·18-s − 1.29e3·19-s − 3.72e3·21-s − 615.·22-s + 4.99e3·23-s − 4.53e3·24-s + 1.56e3·26-s − 1.10e3·27-s + 8.57e3·28-s + 4.10e3·29-s − 4.68e3·31-s − 5.28e3·32-s + 1.53e3·33-s − 4.95e3·34-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 1.48·3-s + 1.66·4-s − 2.41·6-s + 1.24·7-s + 1.08·8-s + 1.19·9-s − 0.166·11-s − 2.46·12-s + 0.277·13-s + 2.02·14-s + 0.105·16-s − 0.450·17-s + 1.95·18-s − 0.823·19-s − 1.84·21-s − 0.271·22-s + 1.96·23-s − 1.60·24-s + 0.452·26-s − 0.291·27-s + 2.06·28-s + 0.907·29-s − 0.874·31-s − 0.912·32-s + 0.246·33-s − 0.734·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.815691441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.815691441\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 9.23T + 32T^{2} \) |
| 3 | \( 1 + 23.1T + 243T^{2} \) |
| 7 | \( 1 - 161.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 66.6T + 1.61e5T^{2} \) |
| 17 | \( 1 + 536.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.29e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.99e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.10e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.24e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.09e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.51e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.94e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.08e5T + 8.58e9T^{2} \) |
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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
−11.17576441627742743520324701254, −10.63128240017200154756300705650, −8.836325174410618328724401941614, −7.36527022266366120969060231682, −6.42958572078760468150661793223, −5.57323457348725140722766866807, −4.85322317860220775279975606042, −4.18124864770257098922971589058, −2.47676786978252190814802811639, −0.949048235686057195720659497292,
0.949048235686057195720659497292, 2.47676786978252190814802811639, 4.18124864770257098922971589058, 4.85322317860220775279975606042, 5.57323457348725140722766866807, 6.42958572078760468150661793223, 7.36527022266366120969060231682, 8.836325174410618328724401941614, 10.63128240017200154756300705650, 11.17576441627742743520324701254
