| L(s) = 1 | + 2·2-s + 4·4-s − 6·5-s + 8·8-s − 12·10-s + 30·11-s − 53·13-s + 16·16-s − 84·17-s + 97·19-s − 24·20-s + 60·22-s − 84·23-s − 89·25-s − 106·26-s + 180·29-s − 179·31-s + 32·32-s − 168·34-s − 145·37-s + 194·38-s − 48·40-s + 126·41-s − 325·43-s + 120·44-s − 168·46-s − 366·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.536·5-s + 0.353·8-s − 0.379·10-s + 0.822·11-s − 1.13·13-s + 1/4·16-s − 1.19·17-s + 1.17·19-s − 0.268·20-s + 0.581·22-s − 0.761·23-s − 0.711·25-s − 0.799·26-s + 1.15·29-s − 1.03·31-s + 0.176·32-s − 0.847·34-s − 0.644·37-s + 0.828·38-s − 0.189·40-s + 0.479·41-s − 1.15·43-s + 0.411·44-s − 0.538·46-s − 1.13·47-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 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 53 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 97 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 180 T + p^{3} T^{2} \) |
| 31 | \( 1 + 179 T + p^{3} T^{2} \) |
| 37 | \( 1 + 145 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 325 T + p^{3} T^{2} \) |
| 47 | \( 1 + 366 T + p^{3} T^{2} \) |
| 53 | \( 1 - 768 T + p^{3} T^{2} \) |
| 59 | \( 1 + 264 T + p^{3} T^{2} \) |
| 61 | \( 1 + 818 T + p^{3} T^{2} \) |
| 67 | \( 1 + 523 T + p^{3} T^{2} \) |
| 71 | \( 1 - 342 T + p^{3} T^{2} \) |
| 73 | \( 1 - 43 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1171 T + p^{3} T^{2} \) |
| 83 | \( 1 + 810 T + p^{3} T^{2} \) |
| 89 | \( 1 + 600 T + p^{3} T^{2} \) |
| 97 | \( 1 + 386 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.406308453863718984307648038112, −8.396152378248967263989390537849, −7.39809039304993864921152605091, −6.79730819980925934687594544970, −5.73914370048033009199460883010, −4.72358279833851795387674357254, −3.98193828550859003805359685159, −2.92221795207241335196625106097, −1.69648999331965213667733923175, 0,
1.69648999331965213667733923175, 2.92221795207241335196625106097, 3.98193828550859003805359685159, 4.72358279833851795387674357254, 5.73914370048033009199460883010, 6.79730819980925934687594544970, 7.39809039304993864921152605091, 8.396152378248967263989390537849, 9.406308453863718984307648038112
