| L(s) = 1 | − 3-s + 3·5-s − 2·7-s − 2·9-s − 4·13-s − 3·15-s − 6·17-s + 8·19-s + 2·21-s − 3·23-s + 4·25-s + 5·27-s + 5·31-s − 6·35-s + 37-s + 4·39-s − 10·43-s − 6·45-s − 3·49-s + 6·51-s + 6·53-s − 8·57-s − 3·59-s − 4·61-s + 4·63-s − 12·65-s + 67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.755·7-s − 2/3·9-s − 1.10·13-s − 0.774·15-s − 1.45·17-s + 1.83·19-s + 0.436·21-s − 0.625·23-s + 4/5·25-s + 0.962·27-s + 0.898·31-s − 1.01·35-s + 0.164·37-s + 0.640·39-s − 1.52·43-s − 0.894·45-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 1.05·57-s − 0.390·59-s − 0.512·61-s + 0.503·63-s − 1.48·65-s + 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.302259704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302259704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.81155080580315786917378596623, −6.82844934181119533069794600545, −6.49239866705743598917256760745, −5.75548364451957323807088643042, −5.21944025828158481102830855776, −4.59295958580532936968902922833, −3.31009578098234720920937956980, −2.63126729182642460425011649857, −1.88312302784001779093982374014, −0.56096039689092099888443756833,
0.56096039689092099888443756833, 1.88312302784001779093982374014, 2.63126729182642460425011649857, 3.31009578098234720920937956980, 4.59295958580532936968902922833, 5.21944025828158481102830855776, 5.75548364451957323807088643042, 6.49239866705743598917256760745, 6.82844934181119533069794600545, 7.81155080580315786917378596623
