L(s) = 1 | − 4·7-s + 4·11-s + 2·17-s − 19-s − 2·23-s − 5·25-s + 6·29-s − 6·31-s − 8·37-s − 10·41-s + 12·43-s + 10·47-s + 9·49-s − 2·53-s + 4·59-s − 10·61-s − 16·71-s − 2·73-s − 16·77-s − 10·79-s − 16·83-s + 2·89-s − 10·97-s − 8·101-s + 6·103-s − 4·107-s − 4·109-s + ⋯ |
L(s) = 1 | − 1.51·7-s + 1.20·11-s + 0.485·17-s − 0.229·19-s − 0.417·23-s − 25-s + 1.11·29-s − 1.07·31-s − 1.31·37-s − 1.56·41-s + 1.82·43-s + 1.45·47-s + 9/7·49-s − 0.274·53-s + 0.520·59-s − 1.28·61-s − 1.89·71-s − 0.234·73-s − 1.82·77-s − 1.12·79-s − 1.75·83-s + 0.211·89-s − 1.01·97-s − 0.796·101-s + 0.591·103-s − 0.386·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.715294333706982997307794001780, −7.51755151440903447029301024156, −6.88215375010934771382988421595, −6.16408975842895416631701933589, −5.58432950026264512671358011636, −4.23486684823494032652708156754, −3.64520587239877913390442824120, −2.77693174390586113318511086678, −1.47847032763559879728730128240, 0,
1.47847032763559879728730128240, 2.77693174390586113318511086678, 3.64520587239877913390442824120, 4.23486684823494032652708156754, 5.58432950026264512671358011636, 6.16408975842895416631701933589, 6.88215375010934771382988421595, 7.51755151440903447029301024156, 8.715294333706982997307794001780