| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 2·13-s + 15-s − 2·19-s − 21-s + 25-s + 5·27-s − 6·29-s + 4·31-s − 35-s − 4·37-s − 2·39-s + 9·41-s + 43-s + 2·45-s + 3·47-s − 6·49-s − 6·53-s + 2·57-s − 61-s − 2·63-s − 2·65-s + 13·67-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.554·13-s + 0.258·15-s − 0.458·19-s − 0.218·21-s + 1/5·25-s + 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.657·37-s − 0.320·39-s + 1.40·41-s + 0.152·43-s + 0.298·45-s + 0.437·47-s − 6/7·49-s − 0.824·53-s + 0.264·57-s − 0.128·61-s − 0.251·63-s − 0.248·65-s + 1.58·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.34598200989106656065915332759, −6.56109353846840068344450922887, −5.96336276249364654963953135115, −5.33936239567196716193229758197, −4.60789967319745721056751104314, −3.88485220425768169549601180433, −3.08050832716075095898475892533, −2.15468709002133937953462276075, −1.06014380128944644954719666061, 0,
1.06014380128944644954719666061, 2.15468709002133937953462276075, 3.08050832716075095898475892533, 3.88485220425768169549601180433, 4.60789967319745721056751104314, 5.33936239567196716193229758197, 5.96336276249364654963953135115, 6.56109353846840068344450922887, 7.34598200989106656065915332759
