L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s + 4·13-s + 16-s − 19-s − 2·22-s + 8·23-s + 4·26-s + 2·29-s − 2·31-s + 32-s + 8·37-s − 38-s + 2·41-s − 4·43-s − 2·44-s + 8·46-s − 4·47-s − 7·49-s + 4·52-s + 2·53-s + 2·58-s − 10·61-s − 2·62-s + 64-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 1.10·13-s + 1/4·16-s − 0.229·19-s − 0.426·22-s + 1.66·23-s + 0.784·26-s + 0.371·29-s − 0.359·31-s + 0.176·32-s + 1.31·37-s − 0.162·38-s + 0.312·41-s − 0.609·43-s − 0.301·44-s + 1.17·46-s − 0.583·47-s − 49-s + 0.554·52-s + 0.274·53-s + 0.262·58-s − 1.28·61-s − 0.254·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.559237367\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.559237367\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.79021055473312040928306723523, −6.89150959582283333248377337529, −6.38824140874180661220044959480, −5.65228618405862476445537804572, −4.96886928560685313828578133355, −4.33947558388033985138440510828, −3.40986908537123631817372725992, −2.88774803558564267062968113783, −1.86579712205530766979031024916, −0.846304249658715054953947633194,
0.846304249658715054953947633194, 1.86579712205530766979031024916, 2.88774803558564267062968113783, 3.40986908537123631817372725992, 4.33947558388033985138440510828, 4.96886928560685313828578133355, 5.65228618405862476445537804572, 6.38824140874180661220044959480, 6.89150959582283333248377337529, 7.79021055473312040928306723523