L(s) = 1 | − 5-s + 7-s − 4·11-s − 2·13-s + 6·17-s + 4·19-s − 8·23-s + 25-s + 2·29-s − 35-s − 2·37-s − 10·41-s + 4·43-s + 49-s − 14·53-s + 4·55-s − 12·59-s − 2·61-s + 2·65-s − 4·67-s + 2·73-s − 4·77-s − 8·79-s + 4·83-s − 6·85-s + 6·89-s − 2·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s − 0.169·35-s − 0.328·37-s − 1.56·41-s + 0.609·43-s + 1/7·49-s − 1.92·53-s + 0.539·55-s − 1.56·59-s − 0.256·61-s + 0.248·65-s − 0.488·67-s + 0.234·73-s − 0.455·77-s − 0.900·79-s + 0.439·83-s − 0.650·85-s + 0.635·89-s − 0.209·91-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 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.186340538781726016557042080569, −7.87901841672678094160195531126, −7.26292608577884900021895978904, −6.09461050051665303981293661244, −5.30826714978461027129562198017, −4.68284985193587084956676888020, −3.55144498368018629340553820634, −2.77303766765864437998898397760, −1.53669394220569060479899859091, 0,
1.53669394220569060479899859091, 2.77303766765864437998898397760, 3.55144498368018629340553820634, 4.68284985193587084956676888020, 5.30826714978461027129562198017, 6.09461050051665303981293661244, 7.26292608577884900021895978904, 7.87901841672678094160195531126, 8.186340538781726016557042080569