L(s) = 1 | + 5-s + 11-s − 5·13-s − 17-s − 6·19-s + 4·23-s + 25-s − 3·29-s + 2·31-s + 8·37-s + 10·41-s − 2·43-s + 7·47-s + 2·53-s + 55-s − 14·59-s − 8·61-s − 5·65-s + 14·67-s − 10·73-s − 11·79-s + 4·83-s − 85-s − 4·89-s − 6·95-s − 3·97-s + 10·101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.301·11-s − 1.38·13-s − 0.242·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.557·29-s + 0.359·31-s + 1.31·37-s + 1.56·41-s − 0.304·43-s + 1.02·47-s + 0.274·53-s + 0.134·55-s − 1.82·59-s − 1.02·61-s − 0.620·65-s + 1.71·67-s − 1.17·73-s − 1.23·79-s + 0.439·83-s − 0.108·85-s − 0.423·89-s − 0.615·95-s − 0.304·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.42052998184274394142801536929, −6.68724481674258146100544226835, −6.10277771644170705491739219098, −5.34369979206434389002021298756, −4.55148392022251346190535085799, −4.05920228036371823861006813414, −2.78069699491447487897997639960, −2.36853818917502493017883032995, −1.28148800407205744453696638855, 0,
1.28148800407205744453696638855, 2.36853818917502493017883032995, 2.78069699491447487897997639960, 4.05920228036371823861006813414, 4.55148392022251346190535085799, 5.34369979206434389002021298756, 6.10277771644170705491739219098, 6.68724481674258146100544226835, 7.42052998184274394142801536929