| L(s) = 1 | − 4.03·3-s + 15.2·5-s + 33.3·7-s − 10.7·9-s − 4.44·11-s − 38.9·13-s − 61.5·15-s − 18.1·17-s + 75.1·19-s − 134.·21-s + 187.·23-s + 107.·25-s + 152.·27-s + 29·29-s + 68.3·31-s + 17.9·33-s + 508.·35-s + 44.9·37-s + 156.·39-s + 299.·41-s − 138.·43-s − 163.·45-s − 531.·47-s + 769.·49-s + 73.0·51-s − 242.·53-s − 67.8·55-s + ⋯ |
| L(s) = 1 | − 0.775·3-s + 1.36·5-s + 1.80·7-s − 0.397·9-s − 0.121·11-s − 0.830·13-s − 1.05·15-s − 0.258·17-s + 0.907·19-s − 1.39·21-s + 1.70·23-s + 0.862·25-s + 1.08·27-s + 0.185·29-s + 0.396·31-s + 0.0945·33-s + 2.45·35-s + 0.199·37-s + 0.644·39-s + 1.13·41-s − 0.490·43-s − 0.542·45-s − 1.64·47-s + 2.24·49-s + 0.200·51-s − 0.629·53-s − 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.002218569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.002218569\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 4.03T + 27T^{2} \) |
| 5 | \( 1 - 15.2T + 125T^{2} \) |
| 7 | \( 1 - 33.3T + 343T^{2} \) |
| 11 | \( 1 + 4.44T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 187.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 68.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 44.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 500.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 325.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 263.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 726.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 851.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 85.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 914.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 662.T + 9.12e5T^{2} \) |
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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
−11.47874631799000630443788236981, −11.00255654791819159762743651529, −9.916323789007765860502761515099, −8.886166442764785015400742726670, −7.73771897866612373701398351916, −6.47595135138267935319286645934, −5.21162443942864300450440860482, −4.99004939028000308705661751612, −2.53079950566105693573706133778, −1.20851519099497757215836712589,
1.20851519099497757215836712589, 2.53079950566105693573706133778, 4.99004939028000308705661751612, 5.21162443942864300450440860482, 6.47595135138267935319286645934, 7.73771897866612373701398351916, 8.886166442764785015400742726670, 9.916323789007765860502761515099, 11.00255654791819159762743651529, 11.47874631799000630443788236981
