| L(s) = 1 | − 5-s − 1.32·11-s − 0.323·13-s + 6.27·17-s − 3.95·19-s − 3.63·23-s + 25-s + 5.23·29-s + 31-s − 4.63·37-s − 8.58·41-s + 5.27·43-s + 3.91·47-s − 12.5·53-s + 1.32·55-s + 7.32·59-s + 2.95·61-s + 0.323·65-s + 0.323·67-s − 4.67·71-s + 3.36·73-s + 10.5·79-s − 8.92·83-s − 6.27·85-s − 13.8·89-s + 3.95·95-s − 3.60·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.398·11-s − 0.0896·13-s + 1.52·17-s − 0.907·19-s − 0.757·23-s + 0.200·25-s + 0.972·29-s + 0.179·31-s − 0.761·37-s − 1.34·41-s + 0.805·43-s + 0.570·47-s − 1.72·53-s + 0.178·55-s + 0.953·59-s + 0.378·61-s + 0.0401·65-s + 0.0394·67-s − 0.555·71-s + 0.394·73-s + 1.18·79-s − 0.979·83-s − 0.681·85-s − 1.47·89-s + 0.405·95-s − 0.365·97-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 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 0.323T + 13T^{2} \) |
| 17 | \( 1 - 6.27T + 17T^{2} \) |
| 19 | \( 1 + 3.95T + 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 + 8.58T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 - 3.91T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.32T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 - 0.323T + 67T^{2} \) |
| 71 | \( 1 + 4.67T + 71T^{2} \) |
| 73 | \( 1 - 3.36T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−7.48635966072515491997448673811, −6.76329268557715266460838779200, −6.04251193623633270422292009996, −5.31501582198471571999853671764, −4.62566021670843276529538235890, −3.80982613714810986485334692932, −3.13745725554918756168225421889, −2.24552915943317034602214269420, −1.18765098519370445792183443683, 0,
1.18765098519370445792183443683, 2.24552915943317034602214269420, 3.13745725554918756168225421889, 3.80982613714810986485334692932, 4.62566021670843276529538235890, 5.31501582198471571999853671764, 6.04251193623633270422292009996, 6.76329268557715266460838779200, 7.48635966072515491997448673811
