| L(s) = 1 | − 0.720·2-s + 1.28·3-s − 7.48·4-s + 3.38·5-s − 0.926·6-s − 21.7·7-s + 11.1·8-s − 25.3·9-s − 2.43·10-s + 50.7·11-s − 9.62·12-s − 65.7·13-s + 15.6·14-s + 4.34·15-s + 51.8·16-s + 3.51·17-s + 18.2·18-s − 25.4·19-s − 25.2·20-s − 27.9·21-s − 36.5·22-s − 96.1·23-s + 14.3·24-s − 113.·25-s + 47.3·26-s − 67.3·27-s + 162.·28-s + ⋯ |
| L(s) = 1 | − 0.254·2-s + 0.247·3-s − 0.935·4-s + 0.302·5-s − 0.0630·6-s − 1.17·7-s + 0.492·8-s − 0.938·9-s − 0.0769·10-s + 1.39·11-s − 0.231·12-s − 1.40·13-s + 0.299·14-s + 0.0748·15-s + 0.809·16-s + 0.0501·17-s + 0.238·18-s − 0.307·19-s − 0.282·20-s − 0.290·21-s − 0.354·22-s − 0.871·23-s + 0.121·24-s − 0.908·25-s + 0.356·26-s − 0.479·27-s + 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7965378050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7965378050\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 0.720T + 8T^{2} \) |
| 3 | \( 1 - 1.28T + 27T^{2} \) |
| 5 | \( 1 - 3.38T + 125T^{2} \) |
| 7 | \( 1 + 21.7T + 343T^{2} \) |
| 11 | \( 1 - 50.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 3.51T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.1T + 1.21e4T^{2} \) |
| 31 | \( 1 - 31.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 320.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 759.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 207.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 459.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 36.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 873.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−9.677054282690273847811392449260, −9.131700569724701857554430084512, −8.350538720341956023002397615265, −7.30147281236896705440377734098, −6.28200807183636859019970070557, −5.49223204167297850088599113701, −4.25120717966905597654083038601, −3.43146220261822028194542723989, −2.18248247453728452519698007781, −0.49033969658310017251829594163,
0.49033969658310017251829594163, 2.18248247453728452519698007781, 3.43146220261822028194542723989, 4.25120717966905597654083038601, 5.49223204167297850088599113701, 6.28200807183636859019970070557, 7.30147281236896705440377734098, 8.350538720341956023002397615265, 9.131700569724701857554430084512, 9.677054282690273847811392449260
