| L(s) = 1 | − 27.0·3-s − 250.·7-s + 491.·9-s + 318.·11-s + 1.07e3·13-s + 1.12e3·17-s + 749.·19-s + 6.79e3·21-s + 250.·23-s − 6.72e3·27-s − 5.62e3·29-s − 7.57e3·31-s − 8.61e3·33-s − 3.07e3·37-s − 2.91e4·39-s + 5.39e3·41-s + 2.08e3·43-s + 4.07e3·47-s + 4.61e4·49-s − 3.03e4·51-s − 9.02e3·53-s − 2.03e4·57-s + 2.74e4·59-s − 4.15e4·61-s − 1.23e5·63-s + 2.77e4·67-s − 6.79e3·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.93·7-s + 2.02·9-s + 0.792·11-s + 1.76·13-s + 0.940·17-s + 0.476·19-s + 3.36·21-s + 0.0989·23-s − 1.77·27-s − 1.24·29-s − 1.41·31-s − 1.37·33-s − 0.368·37-s − 3.07·39-s + 0.500·41-s + 0.172·43-s + 0.269·47-s + 2.74·49-s − 1.63·51-s − 0.441·53-s − 0.827·57-s + 1.02·59-s − 1.43·61-s − 3.91·63-s + 0.756·67-s − 0.171·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8112508323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8112508323\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 27.0T + 243T^{2} \) |
| 7 | \( 1 + 250.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 318.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.07e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 749.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 250.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.62e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.08e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.07e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.02e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.59e4T + 8.58e9T^{2} \) |
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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
−9.638427059771902193235002262829, −9.001802110665965849882522647245, −7.38229410215546090337550993922, −6.62996688730469710185857763167, −5.96105813812884441059817611376, −5.53651188504673112174749590814, −3.95951519036797749826887778334, −3.40685427680940646537662754427, −1.38403471539622402740461073812, −0.49749621938924833726006985357,
0.49749621938924833726006985357, 1.38403471539622402740461073812, 3.40685427680940646537662754427, 3.95951519036797749826887778334, 5.53651188504673112174749590814, 5.96105813812884441059817611376, 6.62996688730469710185857763167, 7.38229410215546090337550993922, 9.001802110665965849882522647245, 9.638427059771902193235002262829
