L(s) = 1 | − 6.34·2-s − 22.4·3-s + 8.30·4-s + 25·5-s + 142.·6-s + 214.·7-s + 150.·8-s + 261.·9-s − 158.·10-s + 121·11-s − 186.·12-s − 442.·13-s − 1.36e3·14-s − 561.·15-s − 1.22e3·16-s − 1.31e3·17-s − 1.65e3·18-s + 361·19-s + 207.·20-s − 4.82e3·21-s − 768.·22-s + 2.15e3·23-s − 3.37e3·24-s + 625·25-s + 2.80e3·26-s − 408.·27-s + 1.78e3·28-s + ⋯ |
L(s) = 1 | − 1.12·2-s − 1.44·3-s + 0.259·4-s + 0.447·5-s + 1.61·6-s + 1.65·7-s + 0.831·8-s + 1.07·9-s − 0.501·10-s + 0.301·11-s − 0.373·12-s − 0.726·13-s − 1.86·14-s − 0.644·15-s − 1.19·16-s − 1.10·17-s − 1.20·18-s + 0.229·19-s + 0.116·20-s − 2.38·21-s − 0.338·22-s + 0.849·23-s − 1.19·24-s + 0.200·25-s + 0.815·26-s − 0.107·27-s + 0.430·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8846059162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8846059162\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 + 6.34T + 32T^{2} \) |
| 3 | \( 1 + 22.4T + 243T^{2} \) |
| 7 | \( 1 - 214.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 442.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.31e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 436.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.46e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.53e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.02e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.14e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.25e4T + 8.58e9T^{2} \) |
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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
−9.184760582541025819694003612652, −8.479196575520496415772320039727, −7.54098521315588848970467073835, −6.83177226349728379328082783721, −5.78477195991595516016427400935, −4.72921463158147402812871368778, −4.60149579922444312708928439745, −2.29428285734113012537419210065, −1.30157158670266559592550393246, −0.60540566279666868560835605592,
0.60540566279666868560835605592, 1.30157158670266559592550393246, 2.29428285734113012537419210065, 4.60149579922444312708928439745, 4.72921463158147402812871368778, 5.78477195991595516016427400935, 6.83177226349728379328082783721, 7.54098521315588848970467073835, 8.479196575520496415772320039727, 9.184760582541025819694003612652