L(s) = 1 | + 5-s + 2·7-s + 3·11-s + 2·17-s − 19-s − 2·23-s + 25-s − 3·29-s + 3·31-s + 2·35-s + 5·41-s + 4·43-s + 8·47-s − 3·49-s + 2·53-s + 3·55-s − 3·59-s + 6·61-s + 10·67-s + 15·71-s − 14·73-s + 6·77-s + 8·79-s + 2·85-s + 89-s − 95-s − 16·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.904·11-s + 0.485·17-s − 0.229·19-s − 0.417·23-s + 1/5·25-s − 0.557·29-s + 0.538·31-s + 0.338·35-s + 0.780·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.404·55-s − 0.390·59-s + 0.768·61-s + 1.22·67-s + 1.78·71-s − 1.63·73-s + 0.683·77-s + 0.900·79-s + 0.216·85-s + 0.105·89-s − 0.102·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.735152851\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.735152851\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.063935604426200674224396391889, −7.31889550864010675363904262307, −6.58490919211320444242571076758, −5.86321794082847699930648840902, −5.24111283507903683306557369099, −4.34570066436012550998241994300, −3.74964088890026628300888660028, −2.62708931807601493968371165414, −1.79104930217700629611956524078, −0.905859705519913020792802597602,
0.905859705519913020792802597602, 1.79104930217700629611956524078, 2.62708931807601493968371165414, 3.74964088890026628300888660028, 4.34570066436012550998241994300, 5.24111283507903683306557369099, 5.86321794082847699930648840902, 6.58490919211320444242571076758, 7.31889550864010675363904262307, 8.063935604426200674224396391889