| L(s) = 1 | + 0.445·3-s + 3.04·5-s − 2.80·9-s + 4.40·11-s − 5.71·13-s + 1.35·15-s − 7.54·17-s + 1.39·19-s − 7.18·23-s + 4.29·25-s − 2.58·27-s − 0.692·29-s + 7.13·31-s + 1.96·33-s + 1.66·37-s − 2.54·39-s − 0.506·41-s − 0.850·43-s − 8.54·45-s + 0.158·47-s − 3.35·51-s − 6.41·53-s + 13.4·55-s + 0.621·57-s − 10.9·59-s − 5.70·61-s − 17.4·65-s + ⋯ |
| L(s) = 1 | + 0.256·3-s + 1.36·5-s − 0.933·9-s + 1.32·11-s − 1.58·13-s + 0.350·15-s − 1.82·17-s + 0.320·19-s − 1.49·23-s + 0.859·25-s − 0.496·27-s − 0.128·29-s + 1.28·31-s + 0.341·33-s + 0.273·37-s − 0.407·39-s − 0.0790·41-s − 0.129·43-s − 1.27·45-s + 0.0231·47-s − 0.470·51-s − 0.880·53-s + 1.81·55-s + 0.0822·57-s − 1.42·59-s − 0.730·61-s − 2.16·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.445T + 3T^{2} \) |
| 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + 7.54T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 + 0.692T + 29T^{2} \) |
| 31 | \( 1 - 7.13T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 0.506T + 41T^{2} \) |
| 43 | \( 1 + 0.850T + 43T^{2} \) |
| 47 | \( 1 - 0.158T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 6.18T + 71T^{2} \) |
| 73 | \( 1 + 1.54T + 73T^{2} \) |
| 79 | \( 1 - 7.84T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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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
−7.86904897167746791036834346593, −6.92637201332727030742439503808, −6.26212120081805298917044608026, −5.86919536855812167292018230283, −4.82532816452868473219782717228, −4.25904239691536144260976299444, −2.99694528760584080752432088679, −2.30839537727281054489422546421, −1.65096710036292790076220116616, 0,
1.65096710036292790076220116616, 2.30839537727281054489422546421, 2.99694528760584080752432088679, 4.25904239691536144260976299444, 4.82532816452868473219782717228, 5.86919536855812167292018230283, 6.26212120081805298917044608026, 6.92637201332727030742439503808, 7.86904897167746791036834346593
