L(s) = 1 | + 4·5-s − 6·13-s − 8·17-s + 11·25-s + 4·29-s + 2·37-s + 8·41-s + 4·53-s − 10·61-s − 24·65-s − 6·73-s − 32·85-s − 16·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 1.66·13-s − 1.94·17-s + 11/5·25-s + 0.742·29-s + 0.328·37-s + 1.24·41-s + 0.549·53-s − 1.28·61-s − 2.97·65-s − 0.702·73-s − 3.47·85-s − 1.69·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.35152379815575, −14.89631515667483, −14.29124480067684, −13.90607580557439, −13.40013360357369, −12.85002677827371, −12.49979413956820, −11.74578132502001, −11.07109853833458, −10.49861159125948, −10.05956947532612, −9.466927689256943, −9.129755986414599, −8.574598391947072, −7.697843905293990, −6.988154421225719, −6.608250983392829, −5.941340653256668, −5.441094189492060, −4.599775408302182, −4.455960268382456, −3.067456604777924, −2.359791957629303, −2.185365193665095, −1.182043340658777, 0,
1.182043340658777, 2.185365193665095, 2.359791957629303, 3.067456604777924, 4.455960268382456, 4.599775408302182, 5.441094189492060, 5.941340653256668, 6.608250983392829, 6.988154421225719, 7.697843905293990, 8.574598391947072, 9.129755986414599, 9.466927689256943, 10.05956947532612, 10.49861159125948, 11.07109853833458, 11.74578132502001, 12.49979413956820, 12.85002677827371, 13.40013360357369, 13.90607580557439, 14.29124480067684, 14.89631515667483, 15.35152379815575