| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 4.26·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 8.53·10-s − 60.5·11-s − 12·12-s + 17.6·13-s − 14·14-s − 12.8·15-s + 16·16-s + 76.9·17-s − 18·18-s − 93.2·19-s + 17.0·20-s − 21·21-s + 121.·22-s + 23·23-s + 24·24-s − 106.·25-s − 35.3·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.381·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.269·10-s − 1.65·11-s − 0.288·12-s + 0.377·13-s − 0.267·14-s − 0.220·15-s + 0.250·16-s + 1.09·17-s − 0.235·18-s − 1.12·19-s + 0.190·20-s − 0.218·21-s + 1.17·22-s + 0.208·23-s + 0.204·24-s − 0.854·25-s − 0.266·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.054621508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054621508\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 4.26T + 125T^{2} \) |
| 11 | \( 1 + 60.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 29 | \( 1 - 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 14.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.67T + 5.06e4T^{2} \) |
| 41 | \( 1 - 3.50T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.37T + 7.95e4T^{2} \) |
| 47 | \( 1 - 7.29T + 1.03e5T^{2} \) |
| 53 | \( 1 - 474.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 252.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 259.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 378.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 269.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 35.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 853.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 46.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−9.941788697134233234015791499808, −8.678506010350436562208086784467, −8.032752527612500879496809534964, −7.25085544894381482034168842279, −6.14009674142889507204004098962, −5.50042020275772088465777027083, −4.50086595522603764945098195328, −2.98015309728765666372283921594, −1.89686447100660029928196183284, −0.62553854077056242550844749235,
0.62553854077056242550844749235, 1.89686447100660029928196183284, 2.98015309728765666372283921594, 4.50086595522603764945098195328, 5.50042020275772088465777027083, 6.14009674142889507204004098962, 7.25085544894381482034168842279, 8.032752527612500879496809534964, 8.678506010350436562208086784467, 9.941788697134233234015791499808
