| L(s) = 1 | − 3-s + 5-s − 1.24·7-s + 9-s − 2·11-s − 1.24·13-s − 15-s − 5.95·17-s − 4·19-s + 1.24·21-s − 5.95·23-s + 25-s − 27-s + 9.20·29-s + 31-s + 2·33-s − 1.24·35-s − 3.24·37-s + 1.24·39-s + 4·41-s − 2·43-s + 45-s − 10.4·47-s − 5.44·49-s + 5.95·51-s − 8.44·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.470·7-s + 0.333·9-s − 0.603·11-s − 0.345·13-s − 0.258·15-s − 1.44·17-s − 0.917·19-s + 0.271·21-s − 1.24·23-s + 0.200·25-s − 0.192·27-s + 1.70·29-s + 0.179·31-s + 0.348·33-s − 0.210·35-s − 0.533·37-s + 0.199·39-s + 0.624·41-s − 0.304·43-s + 0.149·45-s − 1.52·47-s − 0.778·49-s + 0.834·51-s − 1.16·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9228627877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9228627877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 + 5.95T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 - 9.20T + 29T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 59 | \( 1 - 3.20T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 - 9.95T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.107293096591862831365712845183, −6.76790176777894508877390193061, −6.64107706109629690496677995046, −5.92092849123070117874872365641, −4.99622388683140814992154213904, −4.54633828090479290675596983379, −3.59199644764385345733844506946, −2.51586661301099954946807096985, −1.92397151723001513323494834672, −0.47222556087413309400018602582,
0.47222556087413309400018602582, 1.92397151723001513323494834672, 2.51586661301099954946807096985, 3.59199644764385345733844506946, 4.54633828090479290675596983379, 4.99622388683140814992154213904, 5.92092849123070117874872365641, 6.64107706109629690496677995046, 6.76790176777894508877390193061, 8.107293096591862831365712845183
