L(s) = 1 | + 5-s − 7-s + 4·11-s − 6·13-s + 2·17-s − 8·19-s − 4·23-s + 25-s − 6·29-s + 4·31-s − 35-s − 2·37-s + 2·41-s − 12·43-s + 49-s − 2·53-s + 4·55-s + 4·59-s + 6·61-s − 6·65-s − 4·67-s − 8·71-s + 6·73-s − 4·77-s − 16·79-s + 4·83-s + 2·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.20·11-s − 1.66·13-s + 0.485·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s + 0.312·41-s − 1.82·43-s + 1/7·49-s − 0.274·53-s + 0.539·55-s + 0.520·59-s + 0.768·61-s − 0.744·65-s − 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.455·77-s − 1.80·79-s + 0.439·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.622140904396719623567268847879, −7.75504922679689733526129774907, −6.82457844835651829937406499450, −6.35764416632480709325054896567, −5.42979713341541103241638389123, −4.50154877127520804472433685748, −3.72240361617693861622462265493, −2.54115494961707505080689479780, −1.69266759276639637901572331999, 0,
1.69266759276639637901572331999, 2.54115494961707505080689479780, 3.72240361617693861622462265493, 4.50154877127520804472433685748, 5.42979713341541103241638389123, 6.35764416632480709325054896567, 6.82457844835651829937406499450, 7.75504922679689733526129774907, 8.622140904396719623567268847879