| L(s) = 1 | − 1.46·2-s − 5.85·4-s + 9.85·5-s − 29.9·7-s + 20.3·8-s − 14.4·10-s + 46.9·11-s + 43.8·14-s + 17.0·16-s + 48.2·17-s − 120.·19-s − 57.6·20-s − 68.7·22-s − 130.·23-s − 27.9·25-s + 175.·28-s + 194.·29-s + 32.0·31-s − 187.·32-s − 70.7·34-s − 294.·35-s + 32.4·37-s + 176.·38-s + 200.·40-s − 241.·41-s + 96.4·43-s − 274.·44-s + ⋯ |
| L(s) = 1 | − 0.518·2-s − 0.731·4-s + 0.881·5-s − 1.61·7-s + 0.897·8-s − 0.456·10-s + 1.28·11-s + 0.837·14-s + 0.266·16-s + 0.688·17-s − 1.45·19-s − 0.644·20-s − 0.666·22-s − 1.18·23-s − 0.223·25-s + 1.18·28-s + 1.24·29-s + 0.185·31-s − 1.03·32-s − 0.356·34-s − 1.42·35-s + 0.144·37-s + 0.752·38-s + 0.790·40-s − 0.921·41-s + 0.341·43-s − 0.940·44-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 | \( 1 + 1.46T + 8T^{2} \) |
| 5 | \( 1 - 9.85T + 125T^{2} \) |
| 7 | \( 1 + 29.9T + 343T^{2} \) |
| 11 | \( 1 - 46.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 48.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 32.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 32.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 539.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 152.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 327.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 98.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 441.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 773.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 150.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 337.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 169.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 214.T + 9.12e5T^{2} \) |
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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
−8.921775044660620593159892941507, −8.171217594136527454123744307537, −6.93034190532144974186698316386, −6.29053387196781046624742943523, −5.64486075152092329768630900112, −4.27943795959935446084970177265, −3.65876349281430030392574919349, −2.35033188474402870840584206709, −1.12927364861316203486330634664, 0,
1.12927364861316203486330634664, 2.35033188474402870840584206709, 3.65876349281430030392574919349, 4.27943795959935446084970177265, 5.64486075152092329768630900112, 6.29053387196781046624742943523, 6.93034190532144974186698316386, 8.171217594136527454123744307537, 8.921775044660620593159892941507
