| L(s) = 1 | − 8·2-s + 55.4·3-s + 64·4-s + 70.1·5-s − 443.·6-s + 944.·7-s − 512·8-s + 882.·9-s − 561.·10-s − 4.31e3·11-s + 3.54e3·12-s − 1.84e3·13-s − 7.55e3·14-s + 3.88e3·15-s + 4.09e3·16-s + 2.73e4·17-s − 7.05e3·18-s − 3.19e4·19-s + 4.49e3·20-s + 5.23e4·21-s + 3.44e4·22-s − 1.87e4·23-s − 2.83e4·24-s − 7.31e4·25-s + 1.47e4·26-s − 7.22e4·27-s + 6.04e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.18·3-s + 0.5·4-s + 0.251·5-s − 0.837·6-s + 1.04·7-s − 0.353·8-s + 0.403·9-s − 0.177·10-s − 0.976·11-s + 0.592·12-s − 0.232·13-s − 0.736·14-s + 0.297·15-s + 0.250·16-s + 1.34·17-s − 0.285·18-s − 1.06·19-s + 0.125·20-s + 1.23·21-s + 0.690·22-s − 0.321·23-s − 0.418·24-s − 0.936·25-s + 0.164·26-s − 0.706·27-s + 0.520·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.853174188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.853174188\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 - 55.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 70.1T + 7.81e4T^{2} \) |
| 7 | \( 1 - 944.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.31e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.84e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.73e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.87e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.50e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.44e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 6.79e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.72e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.56e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.98e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.86e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.95e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 8.78e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.26e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.72e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.55e6T + 8.07e13T^{2} \) |
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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
−9.616869829508804046323109149613, −8.641226063150122774696958290359, −7.924683956290719094740365429675, −7.64884229222007036185068225347, −6.12750281022285702832856943492, −5.07884344599562627498072180903, −3.79076573830019125709081936015, −2.55715062105100101604525192906, −2.02575210920403935476078143955, −0.75866401028503211450234978664,
0.75866401028503211450234978664, 2.02575210920403935476078143955, 2.55715062105100101604525192906, 3.79076573830019125709081936015, 5.07884344599562627498072180903, 6.12750281022285702832856943492, 7.64884229222007036185068225347, 7.924683956290719094740365429675, 8.641226063150122774696958290359, 9.616869829508804046323109149613
