| L(s) = 1 | + 3.05·2-s + 3·3-s + 1.30·4-s − 6.41·5-s + 9.15·6-s + 13.3·7-s − 20.4·8-s + 9·9-s − 19.5·10-s − 11·11-s + 3.92·12-s − 14.1·13-s + 40.6·14-s − 19.2·15-s − 72.7·16-s + 25.9·17-s + 27.4·18-s + 149.·19-s − 8.40·20-s + 39.9·21-s − 33.5·22-s − 44.5·23-s − 61.2·24-s − 83.7·25-s − 43.1·26-s + 27·27-s + 17.4·28-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.577·3-s + 0.163·4-s − 0.574·5-s + 0.622·6-s + 0.719·7-s − 0.902·8-s + 0.333·9-s − 0.619·10-s − 0.301·11-s + 0.0944·12-s − 0.302·13-s + 0.776·14-s − 0.331·15-s − 1.13·16-s + 0.370·17-s + 0.359·18-s + 1.80·19-s − 0.0939·20-s + 0.415·21-s − 0.325·22-s − 0.403·23-s − 0.520·24-s − 0.670·25-s − 0.325·26-s + 0.192·27-s + 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.097827601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.097827601\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 3.05T + 8T^{2} \) |
| 5 | \( 1 + 6.41T + 125T^{2} \) |
| 7 | \( 1 - 13.3T + 343T^{2} \) |
| 13 | \( 1 + 14.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 149.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 283.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 70.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 75.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 224.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 377.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 298.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 753.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 566.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 22.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 870.T + 9.12e5T^{2} \) |
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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
−8.553338230286087292264632904851, −8.052183332687249552610542416653, −7.27842786290346630903952678729, −6.31253809385711657559985424671, −5.14510425049609673126594426117, −4.88608211485663323831434906776, −3.76081121099927283663598398215, −3.23599241014658983663456051061, −2.18056640179247292876508306055, −0.77002720281945495961400659815,
0.77002720281945495961400659815, 2.18056640179247292876508306055, 3.23599241014658983663456051061, 3.76081121099927283663598398215, 4.88608211485663323831434906776, 5.14510425049609673126594426117, 6.31253809385711657559985424671, 7.27842786290346630903952678729, 8.052183332687249552610542416653, 8.553338230286087292264632904851
