| L(s) = 1 | − 2·2-s + 4·4-s − 7.41·5-s − 8·8-s + 14.8·10-s − 10.4·11-s + 2.78·13-s + 16·16-s − 50.4·17-s + 125.·19-s − 29.6·20-s + 20.9·22-s + 182.·23-s − 70.0·25-s − 5.57·26-s − 156.·29-s + 139.·31-s − 32·32-s + 100.·34-s − 394.·37-s − 250.·38-s + 59.3·40-s − 197.·41-s + 343.·43-s − 41.9·44-s − 364.·46-s + 610.·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.663·5-s − 0.353·8-s + 0.468·10-s − 0.287·11-s + 0.0594·13-s + 0.250·16-s − 0.719·17-s + 1.50·19-s − 0.331·20-s + 0.203·22-s + 1.65·23-s − 0.560·25-s − 0.0420·26-s − 0.999·29-s + 0.808·31-s − 0.176·32-s + 0.508·34-s − 1.75·37-s − 1.06·38-s + 0.234·40-s − 0.752·41-s + 1.21·43-s − 0.143·44-s − 1.16·46-s + 1.89·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 7.41T + 125T^{2} \) |
| 11 | \( 1 + 10.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.78T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 394.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 610.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 247.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 997.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 848.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 553.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 903.T + 9.12e5T^{2} \) |
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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
−9.155660751676870838734898390465, −8.649401809578329842333574905679, −7.43968822219664070941235706518, −7.23561181822408275497593318828, −5.90801905790919284798576446434, −4.92393863039880438068452496732, −3.68945485053633272072374993435, −2.68632041969242410553035852191, −1.25251076799397200600005695343, 0,
1.25251076799397200600005695343, 2.68632041969242410553035852191, 3.68945485053633272072374993435, 4.92393863039880438068452496732, 5.90801905790919284798576446434, 7.23561181822408275497593318828, 7.43968822219664070941235706518, 8.649401809578329842333574905679, 9.155660751676870838734898390465
