| L(s) = 1 | − 3-s + 2.64·5-s + 0.484·7-s + 9-s − 2.48·11-s + 0.484·13-s − 2.64·15-s − 7.37·17-s − 7.76·19-s − 0.484·21-s − 7.12·23-s + 1.96·25-s − 27-s − 2.64·29-s + 6.24·31-s + 2.48·33-s + 1.28·35-s − 37-s − 0.484·39-s + 2·41-s − 7.21·43-s + 2.64·45-s + 7.52·47-s − 6.76·49-s + 7.37·51-s + 12.9·53-s − 6.56·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.18·5-s + 0.183·7-s + 0.333·9-s − 0.749·11-s + 0.134·13-s − 0.681·15-s − 1.78·17-s − 1.78·19-s − 0.105·21-s − 1.48·23-s + 0.393·25-s − 0.192·27-s − 0.490·29-s + 1.12·31-s + 0.432·33-s + 0.216·35-s − 0.164·37-s − 0.0776·39-s + 0.312·41-s − 1.10·43-s + 0.393·45-s + 1.09·47-s − 0.966·49-s + 1.03·51-s + 1.78·53-s − 0.884·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 - 0.484T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 0.484T + 13T^{2} \) |
| 17 | \( 1 + 7.37T + 17T^{2} \) |
| 19 | \( 1 + 7.76T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 + 2.64T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 7.52T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 9.70T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 9.43T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.851455698461039698223463439945, −8.280569386037317572083020975426, −7.11351600247246041931699528758, −6.23944400295215953250186986455, −5.89141768001956615136597565088, −4.80136189780041508772355869806, −4.12456085712115910706291318180, −2.44249824334351994188219115185, −1.86281958704487215127875111537, 0,
1.86281958704487215127875111537, 2.44249824334351994188219115185, 4.12456085712115910706291318180, 4.80136189780041508772355869806, 5.89141768001956615136597565088, 6.23944400295215953250186986455, 7.11351600247246041931699528758, 8.280569386037317572083020975426, 8.851455698461039698223463439945
