| L(s) = 1 | + 3.31·2-s − 3·3-s + 3.01·4-s − 6.71·5-s − 9.95·6-s + 13.2·7-s − 16.5·8-s + 9·9-s − 22.2·10-s + 11·11-s − 9.04·12-s + 5.36·13-s + 44.1·14-s + 20.1·15-s − 79.0·16-s − 5.77·17-s + 29.8·18-s + 58.0·19-s − 20.2·20-s − 39.8·21-s + 36.5·22-s − 32.8·23-s + 49.6·24-s − 79.9·25-s + 17.8·26-s − 27·27-s + 40.0·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s − 0.577·3-s + 0.376·4-s − 0.600·5-s − 0.677·6-s + 0.717·7-s − 0.731·8-s + 0.333·9-s − 0.704·10-s + 0.301·11-s − 0.217·12-s + 0.114·13-s + 0.842·14-s + 0.346·15-s − 1.23·16-s − 0.0823·17-s + 0.391·18-s + 0.701·19-s − 0.226·20-s − 0.414·21-s + 0.353·22-s − 0.298·23-s + 0.422·24-s − 0.639·25-s + 0.134·26-s − 0.192·27-s + 0.270·28-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
| \(L(2)\) |
\(\approx\) |
\(2.488441077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488441077\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.31T + 8T^{2} \) |
| 5 | \( 1 + 6.71T + 125T^{2} \) |
| 7 | \( 1 - 13.2T + 343T^{2} \) |
| 13 | \( 1 - 5.36T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.77T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 63.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 221.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 140.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 278.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 232.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 214.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 73.0T + 2.05e5T^{2} \) |
| 67 | \( 1 + 832.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 236.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 278.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 940.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 259.T + 9.12e5T^{2} \) |
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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
−8.806339554571847584801821577406, −7.78746209504973350306569383537, −7.13896409184044204417456893579, −6.07335317639073689274133512329, −5.52746138874554679580248368219, −4.66872253015759120980871608180, −4.07226733741404342341882409026, −3.27517118533471916059605123534, −1.97590723343589131459675826377, −0.62116264493802659338311778861,
0.62116264493802659338311778861, 1.97590723343589131459675826377, 3.27517118533471916059605123534, 4.07226733741404342341882409026, 4.66872253015759120980871608180, 5.52746138874554679580248368219, 6.07335317639073689274133512329, 7.13896409184044204417456893579, 7.78746209504973350306569383537, 8.806339554571847584801821577406
