| L(s) = 1 | − 2-s + 3.41·3-s + 4-s + 1.24·5-s − 3.41·6-s + 3.09·7-s − 8-s + 8.66·9-s − 1.24·10-s − 0.904·11-s + 3.41·12-s − 5.39·13-s − 3.09·14-s + 4.24·15-s + 16-s − 0.870·17-s − 8.66·18-s − 3.19·19-s + 1.24·20-s + 10.5·21-s + 0.904·22-s + 8.26·23-s − 3.41·24-s − 3.45·25-s + 5.39·26-s + 19.3·27-s + 3.09·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.97·3-s + 0.5·4-s + 0.556·5-s − 1.39·6-s + 1.17·7-s − 0.353·8-s + 2.88·9-s − 0.393·10-s − 0.272·11-s + 0.985·12-s − 1.49·13-s − 0.828·14-s + 1.09·15-s + 0.250·16-s − 0.211·17-s − 2.04·18-s − 0.733·19-s + 0.278·20-s + 2.30·21-s + 0.192·22-s + 1.72·23-s − 0.697·24-s − 0.690·25-s + 1.05·26-s + 3.72·27-s + 0.585·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.007613311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.007613311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 - 3.09T + 7T^{2} \) |
| 11 | \( 1 + 0.904T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 0.870T + 17T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 - 8.26T + 23T^{2} \) |
| 29 | \( 1 + 4.98T + 29T^{2} \) |
| 31 | \( 1 - 9.63T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 43 | \( 1 + 2.07T + 43T^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 + 9.37T + 53T^{2} \) |
| 59 | \( 1 - 7.32T + 59T^{2} \) |
| 61 | \( 1 + 0.973T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 5.80T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 0.715T + 97T^{2} \) |
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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
−9.452081036529460019293546272753, −8.582001128404320201639127215902, −8.104643306611769578078236759516, −7.40919322157570932780350241929, −6.73879674404268693522561101806, −5.10257856995714959039441693945, −4.34512220111129949525861567536, −2.96920995374099070567041239489, −2.28084472553717757711340036843, −1.51409937890078978088880096012,
1.51409937890078978088880096012, 2.28084472553717757711340036843, 2.96920995374099070567041239489, 4.34512220111129949525861567536, 5.10257856995714959039441693945, 6.73879674404268693522561101806, 7.40919322157570932780350241929, 8.104643306611769578078236759516, 8.582001128404320201639127215902, 9.452081036529460019293546272753
