| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 4.32·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 8.65·10-s − 69.5·11-s + 12·12-s + 90.0·13-s − 14·14-s + 12.9·15-s + 16·16-s − 107.·17-s − 18·18-s + 97.7·19-s + 17.3·20-s + 21·21-s + 139.·22-s − 23·23-s − 24·24-s − 106.·25-s − 180.·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.386·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.273·10-s − 1.90·11-s + 0.288·12-s + 1.92·13-s − 0.267·14-s + 0.223·15-s + 0.250·16-s − 1.52·17-s − 0.235·18-s + 1.18·19-s + 0.193·20-s + 0.218·21-s + 1.34·22-s − 0.208·23-s − 0.204·24-s − 0.850·25-s − 1.35·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.020371290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020371290\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 4.32T + 125T^{2} \) |
| 11 | \( 1 + 69.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 88.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 246.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 84.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 644.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 41.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 541.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 439.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 579.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 344.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 278.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 586.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−9.568707599569711746258734526085, −8.651929271045745534929882289764, −8.173747590402695098045258953705, −7.39534402988606949642907384061, −6.28188300218349236337114851431, −5.44870551544932578290183926278, −4.21327935580634819772400631602, −2.91969449820132926103350496859, −2.09750678731962252441728536389, −0.832598504287769009104908513529,
0.832598504287769009104908513529, 2.09750678731962252441728536389, 2.91969449820132926103350496859, 4.21327935580634819772400631602, 5.44870551544932578290183926278, 6.28188300218349236337114851431, 7.39534402988606949642907384061, 8.173747590402695098045258953705, 8.651929271045745534929882289764, 9.568707599569711746258734526085
