| L(s) = 1 | + 10.1·2-s + 28.8·3-s + 71.2·4-s − 25·5-s + 293.·6-s − 182.·7-s + 399.·8-s + 590.·9-s − 254.·10-s + 121·11-s + 2.05e3·12-s + 363.·13-s − 1.85e3·14-s − 721.·15-s + 1.77e3·16-s + 1.89e3·17-s + 5.99e3·18-s − 361·19-s − 1.78e3·20-s − 5.26e3·21-s + 1.22e3·22-s + 2.12e3·23-s + 1.15e4·24-s + 625·25-s + 3.69e3·26-s + 1.00e4·27-s − 1.30e4·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 1.85·3-s + 2.22·4-s − 0.447·5-s + 3.32·6-s − 1.40·7-s + 2.20·8-s + 2.42·9-s − 0.803·10-s + 0.301·11-s + 4.12·12-s + 0.596·13-s − 2.52·14-s − 0.828·15-s + 1.73·16-s + 1.58·17-s + 4.36·18-s − 0.229·19-s − 0.996·20-s − 2.60·21-s + 0.541·22-s + 0.835·23-s + 4.08·24-s + 0.200·25-s + 1.07·26-s + 2.64·27-s − 3.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(14.22051796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.22051796\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 10.1T + 32T^{2} \) |
| 3 | \( 1 - 28.8T + 243T^{2} \) |
| 7 | \( 1 + 182.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 363.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.89e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.42e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.49e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 589.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.89e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.32e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.69e4T + 8.58e9T^{2} \) |
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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
−9.143416303172706176883039716027, −8.227559635876943241172332586023, −7.25237102408090314028113244170, −6.72828974936961016258458349929, −5.68611935254621507565971758388, −4.42852328308717132793615041844, −3.56928599656304904840119984781, −3.28777716667019352788340980192, −2.54273100892508400175907004799, −1.24843523542861121176678657342,
1.24843523542861121176678657342, 2.54273100892508400175907004799, 3.28777716667019352788340980192, 3.56928599656304904840119984781, 4.42852328308717132793615041844, 5.68611935254621507565971758388, 6.72828974936961016258458349929, 7.25237102408090314028113244170, 8.227559635876943241172332586023, 9.143416303172706176883039716027
