L(s) = 1 | − 3-s − 3·7-s − 2·9-s + 2·11-s − 13-s + 5·17-s + 19-s + 3·21-s + 23-s + 5·27-s − 3·29-s + 4·31-s − 2·33-s − 2·37-s + 39-s − 8·41-s + 8·43-s + 8·47-s + 2·49-s − 5·51-s − 9·53-s − 57-s + 59-s + 14·61-s + 6·63-s − 13·67-s − 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s + 0.603·11-s − 0.277·13-s + 1.21·17-s + 0.229·19-s + 0.654·21-s + 0.208·23-s + 0.962·27-s − 0.557·29-s + 0.718·31-s − 0.348·33-s − 0.328·37-s + 0.160·39-s − 1.24·41-s + 1.21·43-s + 1.16·47-s + 2/7·49-s − 0.700·51-s − 1.23·53-s − 0.132·57-s + 0.130·59-s + 1.79·61-s + 0.755·63-s − 1.58·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.166648267265982398111566767460, −7.20310311884136833953679284959, −6.62672317230234292498020652502, −5.81320435637810296582642337516, −5.39876413354883715323509552770, −4.26995494061736035952806129372, −3.35294656451373150324017695568, −2.70735587597500828235274945486, −1.20593462762231457794198400382, 0,
1.20593462762231457794198400382, 2.70735587597500828235274945486, 3.35294656451373150324017695568, 4.26995494061736035952806129372, 5.39876413354883715323509552770, 5.81320435637810296582642337516, 6.62672317230234292498020652502, 7.20310311884136833953679284959, 8.166648267265982398111566767460