| L(s) = 1 | − 4·2-s + 16·4-s + 75.4·5-s − 64·8-s − 301.·10-s + 149.·11-s + 349.·13-s + 256·16-s + 1.14e3·17-s − 2.79e3·19-s + 1.20e3·20-s − 597.·22-s − 1.81e3·23-s + 2.57e3·25-s − 1.39e3·26-s + 759.·29-s + 9.03e3·31-s − 1.02e3·32-s − 4.59e3·34-s + 7.79e3·37-s + 1.11e4·38-s − 4.83e3·40-s − 7.64e3·41-s + 1.21e4·43-s + 2.39e3·44-s + 7.25e3·46-s − 2.45e4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.35·5-s − 0.353·8-s − 0.954·10-s + 0.372·11-s + 0.573·13-s + 0.250·16-s + 0.964·17-s − 1.77·19-s + 0.675·20-s − 0.263·22-s − 0.715·23-s + 0.823·25-s − 0.405·26-s + 0.167·29-s + 1.68·31-s − 0.176·32-s − 0.682·34-s + 0.936·37-s + 1.25·38-s − 0.477·40-s − 0.709·41-s + 1.00·43-s + 0.186·44-s + 0.505·46-s − 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.327936979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.327936979\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 75.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 149.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 349.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.81e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 759.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.23e4T + 8.58e9T^{2} \) |
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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
−9.593686679517239067262318382131, −8.544752750726073573826070796095, −7.998780579032255700507522745134, −6.47022427341208347137695788168, −6.33179203700402331810076632712, −5.21970523094121470338683659920, −3.92487015276632896605164943347, −2.58498643910345095811980946115, −1.76631922511798681376834421417, −0.78171304658874428420891550419,
0.78171304658874428420891550419, 1.76631922511798681376834421417, 2.58498643910345095811980946115, 3.92487015276632896605164943347, 5.21970523094121470338683659920, 6.33179203700402331810076632712, 6.47022427341208347137695788168, 7.998780579032255700507522745134, 8.544752750726073573826070796095, 9.593686679517239067262318382131
