| L(s) = 1 | + 13.1·3-s + 1.54·5-s − 82.4·7-s − 69.2·9-s − 22.4·11-s + 1.10e3·13-s + 20.3·15-s − 841.·17-s − 941.·19-s − 1.08e3·21-s + 2.35e3·23-s − 3.12e3·25-s − 4.11e3·27-s − 351.·29-s + 3.38e3·31-s − 295.·33-s − 127.·35-s + 8.92e3·37-s + 1.46e4·39-s + 36.4·41-s + 1.64e4·43-s − 106.·45-s − 2.05e4·47-s − 1.00e4·49-s − 1.10e4·51-s − 4.04e4·53-s − 34.5·55-s + ⋯ |
| L(s) = 1 | + 0.845·3-s + 0.0275·5-s − 0.636·7-s − 0.285·9-s − 0.0558·11-s + 1.82·13-s + 0.0233·15-s − 0.706·17-s − 0.598·19-s − 0.537·21-s + 0.926·23-s − 0.999·25-s − 1.08·27-s − 0.0776·29-s + 0.632·31-s − 0.0472·33-s − 0.0175·35-s + 1.07·37-s + 1.54·39-s + 0.00338·41-s + 1.35·43-s − 0.00785·45-s − 1.35·47-s − 0.595·49-s − 0.597·51-s − 1.97·53-s − 0.00153·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 89 | \( 1 + 7.92e3T \) |
| good | 3 | \( 1 - 13.1T + 243T^{2} \) |
| 5 | \( 1 - 1.54T + 3.12e3T^{2} \) |
| 7 | \( 1 + 82.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 22.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.10e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 841.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 941.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 351.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 36.4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.64e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.31e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.68e4T + 3.93e9T^{2} \) |
| 97 | \( 1 - 3.75e4T + 8.58e9T^{2} \) |
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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
−9.060528911202466031259191395741, −8.519723210048209836620741792105, −7.69661989063447584156682890443, −6.43773663514499033169266062243, −5.91751454904785359070706271236, −4.40910880840356215942976265175, −3.47329884767616049273240829794, −2.68166545952270857156640473991, −1.44125662498447175648904307799, 0,
1.44125662498447175648904307799, 2.68166545952270857156640473991, 3.47329884767616049273240829794, 4.40910880840356215942976265175, 5.91751454904785359070706271236, 6.43773663514499033169266062243, 7.69661989063447584156682890443, 8.519723210048209836620741792105, 9.060528911202466031259191395741
