| L(s) = 1 | + 2·2-s + 4·4-s − 14·5-s + 8·8-s − 28·10-s + 28·11-s − 18·13-s + 16·16-s + 74·17-s − 80·19-s − 56·20-s + 56·22-s + 112·23-s + 71·25-s − 36·26-s − 190·29-s − 72·31-s + 32·32-s + 148·34-s − 346·37-s − 160·38-s − 112·40-s + 162·41-s − 412·43-s + 112·44-s + 224·46-s + 24·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.25·5-s + 0.353·8-s − 0.885·10-s + 0.767·11-s − 0.384·13-s + 1/4·16-s + 1.05·17-s − 0.965·19-s − 0.626·20-s + 0.542·22-s + 1.01·23-s + 0.567·25-s − 0.271·26-s − 1.21·29-s − 0.417·31-s + 0.176·32-s + 0.746·34-s − 1.53·37-s − 0.683·38-s − 0.442·40-s + 0.617·41-s − 1.46·43-s + 0.383·44-s + 0.717·46-s + 0.0744·47-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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 - 162 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 198 T + p^{3} T^{2} \) |
| 67 | \( 1 + 716 T + p^{3} T^{2} \) |
| 71 | \( 1 + 392 T + p^{3} T^{2} \) |
| 73 | \( 1 + 538 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1354 T + p^{3} T^{2} \) |
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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
−9.265274438162055740245695096757, −8.347041396783567908745419755907, −7.45529307952446536622558046277, −6.85232676770949188464314977804, −5.71375321137227813688202174031, −4.71102207258056269869486629402, −3.84651720561262429955352481426, −3.14044201720012451424000784493, −1.58018493530109301075161790176, 0,
1.58018493530109301075161790176, 3.14044201720012451424000784493, 3.84651720561262429955352481426, 4.71102207258056269869486629402, 5.71375321137227813688202174031, 6.85232676770949188464314977804, 7.45529307952446536622558046277, 8.347041396783567908745419755907, 9.265274438162055740245695096757
