L(s) = 1 | + 3·7-s − 3·9-s − 4·11-s − 2·13-s − 8·17-s + 3·19-s + 3·23-s − 5·25-s + 7·29-s + 10·31-s + 7·37-s + 9·41-s − 43-s + 11·47-s + 2·49-s + 10·53-s + 5·61-s − 9·63-s + 10·67-s − 8·71-s + 6·73-s − 12·77-s − 2·79-s + 9·81-s + 7·83-s − 8·89-s − 6·91-s + ⋯ |
L(s) = 1 | + 1.13·7-s − 9-s − 1.20·11-s − 0.554·13-s − 1.94·17-s + 0.688·19-s + 0.625·23-s − 25-s + 1.29·29-s + 1.79·31-s + 1.15·37-s + 1.40·41-s − 0.152·43-s + 1.60·47-s + 2/7·49-s + 1.37·53-s + 0.640·61-s − 1.13·63-s + 1.22·67-s − 0.949·71-s + 0.702·73-s − 1.36·77-s − 0.225·79-s + 81-s + 0.768·83-s − 0.847·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.597724116\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597724116\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 197 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.503542721403122546808063788687, −8.056603231480438386460344076725, −7.33956567594263867864342447503, −6.37618381971660332853552601773, −5.51910959519898081306299022355, −4.84286071876505944866542752996, −4.21913762037456798202222334655, −2.60609363958670334343605384522, −2.44365137125241217402163994690, −0.74177520741704320779285872897,
0.74177520741704320779285872897, 2.44365137125241217402163994690, 2.60609363958670334343605384522, 4.21913762037456798202222334655, 4.84286071876505944866542752996, 5.51910959519898081306299022355, 6.37618381971660332853552601773, 7.33956567594263867864342447503, 8.056603231480438386460344076725, 8.503542721403122546808063788687