| L(s) = 1 | + 1.05·2-s − 6.89·4-s − 17.8·5-s − 30.1·7-s − 15.6·8-s − 18.8·10-s + 50.8·11-s − 31.7·14-s + 38.6·16-s + 2.99·17-s + 72.7·19-s + 123.·20-s + 53.4·22-s + 41.9·23-s + 195.·25-s + 208.·28-s + 135.·29-s − 316.·31-s + 165.·32-s + 3.14·34-s + 540.·35-s + 261.·37-s + 76.4·38-s + 280.·40-s − 198.·41-s − 201.·43-s − 350.·44-s + ⋯ |
| L(s) = 1 | + 0.371·2-s − 0.861·4-s − 1.60·5-s − 1.63·7-s − 0.692·8-s − 0.594·10-s + 1.39·11-s − 0.606·14-s + 0.604·16-s + 0.0426·17-s + 0.877·19-s + 1.37·20-s + 0.518·22-s + 0.379·23-s + 1.56·25-s + 1.40·28-s + 0.865·29-s − 1.83·31-s + 0.916·32-s + 0.0158·34-s + 2.60·35-s + 1.16·37-s + 0.326·38-s + 1.10·40-s − 0.757·41-s − 0.714·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.05T + 8T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 7 | \( 1 + 30.1T + 343T^{2} \) |
| 11 | \( 1 - 50.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 2.99T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 316.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 97.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 497.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 525.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 777.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 612.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 718.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 397.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 648.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 272.T + 9.12e5T^{2} \) |
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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
−8.861841415688076997648065620265, −7.907629908638405907915981570046, −6.99341240922941176150350669789, −6.35889395964144146267668313478, −5.24527575216636105581535914534, −4.18643729684408348371243036219, −3.61787832365744119953567555673, −3.12200961742351463194891151350, −0.899851130727178962054585234175, 0,
0.899851130727178962054585234175, 3.12200961742351463194891151350, 3.61787832365744119953567555673, 4.18643729684408348371243036219, 5.24527575216636105581535914534, 6.35889395964144146267668313478, 6.99341240922941176150350669789, 7.907629908638405907915981570046, 8.861841415688076997648065620265
