| L(s) = 1 | − 3-s + 2.73·5-s − 2·7-s + 9-s − 5.46·11-s − 3.46·13-s − 2.73·15-s − 2.73·17-s + 1.46·19-s + 2·21-s − 4.73·23-s + 2.46·25-s − 27-s + 4.19·29-s − 4·31-s + 5.46·33-s − 5.46·35-s − 37-s + 3.46·39-s − 2·41-s + 2.92·43-s + 2.73·45-s + 6.92·47-s − 3·49-s + 2.73·51-s − 10.3·53-s − 14.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.22·5-s − 0.755·7-s + 0.333·9-s − 1.64·11-s − 0.960·13-s − 0.705·15-s − 0.662·17-s + 0.335·19-s + 0.436·21-s − 0.986·23-s + 0.492·25-s − 0.192·27-s + 0.779·29-s − 0.718·31-s + 0.951·33-s − 0.923·35-s − 0.164·37-s + 0.554·39-s − 0.312·41-s + 0.446·43-s + 0.407·45-s + 1.01·47-s − 0.428·49-s + 0.382·51-s − 1.42·53-s − 2.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.73T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 + 9.46T + 83T^{2} \) |
| 89 | \( 1 - 8.19T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−9.971378861739795549245201425151, −9.130309226798510467327904748233, −7.906720972154092437548843169167, −7.01417520904193761282666320921, −6.06854642629549587393233711106, −5.46026607136367643119981998992, −4.58021611978117042354028755337, −2.94336062776798121758206614499, −2.03379657439542754093790189426, 0,
2.03379657439542754093790189426, 2.94336062776798121758206614499, 4.58021611978117042354028755337, 5.46026607136367643119981998992, 6.06854642629549587393233711106, 7.01417520904193761282666320921, 7.906720972154092437548843169167, 9.130309226798510467327904748233, 9.971378861739795549245201425151
