| L(s) = 1 | − 3·3-s − 11.7·5-s + 15.4·7-s + 9·9-s + 59.6·13-s + 35.3·15-s + 28.9·17-s − 142.·19-s − 46.4·21-s + 185·23-s + 14.2·25-s − 27·27-s − 155.·29-s + 283.·31-s − 182.·35-s − 202.·37-s − 178.·39-s + 294.·41-s − 179.·43-s − 106.·45-s − 408.·47-s − 103.·49-s − 86.9·51-s + 359.·53-s + 427.·57-s + 288.·59-s − 151.·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.05·5-s + 0.835·7-s + 0.333·9-s + 1.27·13-s + 0.609·15-s + 0.413·17-s − 1.72·19-s − 0.482·21-s + 1.67·23-s + 0.113·25-s − 0.192·27-s − 0.993·29-s + 1.64·31-s − 0.881·35-s − 0.898·37-s − 0.734·39-s + 1.12·41-s − 0.635·43-s − 0.351·45-s − 1.26·47-s − 0.302·49-s − 0.238·51-s + 0.931·53-s + 0.993·57-s + 0.635·59-s − 0.318·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.458622106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458622106\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 11.7T + 125T^{2} \) |
| 7 | \( 1 - 15.4T + 343T^{2} \) |
| 13 | \( 1 - 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185T + 1.21e4T^{2} \) |
| 29 | \( 1 + 155.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 283.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 294.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 179.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 408.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 359.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 288.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 151.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 244.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 562.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 54.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 554.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 144.T + 9.12e5T^{2} \) |
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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
−8.859839819800511609886779653560, −8.373400095474496091276731865400, −7.59356832946126551108051340259, −6.71399591441195312222177453321, −5.89017220128086901323364753356, −4.82402181601525221324812673409, −4.19315555558748163038703205006, −3.24333162580861226190705906985, −1.70792028063166588774811413935, −0.63602624670568092084209928456,
0.63602624670568092084209928456, 1.70792028063166588774811413935, 3.24333162580861226190705906985, 4.19315555558748163038703205006, 4.82402181601525221324812673409, 5.89017220128086901323364753356, 6.71399591441195312222177453321, 7.59356832946126551108051340259, 8.373400095474496091276731865400, 8.859839819800511609886779653560
