| L(s) = 1 | − 2.09e6·2-s − 5.75e9·3-s + 4.39e12·4-s − 8.77e14·5-s + 1.20e16·6-s − 5.58e17·7-s − 9.22e18·8-s − 2.95e20·9-s + 1.84e21·10-s + 2.59e22·11-s − 2.53e22·12-s + 1.25e24·13-s + 1.17e24·14-s + 5.05e24·15-s + 1.93e25·16-s + 1.55e26·17-s + 6.18e26·18-s − 3.27e27·19-s − 3.85e27·20-s + 3.21e27·21-s − 5.43e28·22-s − 9.13e28·23-s + 5.30e28·24-s − 3.66e29·25-s − 2.62e30·26-s + 3.58e30·27-s − 2.45e30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.317·3-s + 0.5·4-s − 0.822·5-s + 0.224·6-s − 0.377·7-s − 0.353·8-s − 0.899·9-s + 0.581·10-s + 1.05·11-s − 0.158·12-s + 1.40·13-s + 0.267·14-s + 0.261·15-s + 0.250·16-s + 0.544·17-s + 0.635·18-s − 1.05·19-s − 0.411·20-s + 0.120·21-s − 0.746·22-s − 0.482·23-s + 0.112·24-s − 0.322·25-s − 0.993·26-s + 0.603·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(0.7376002102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7376002102\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.09e6T \) |
| 7 | \( 1 + 5.58e17T \) |
| good | 3 | \( 1 + 5.75e9T + 3.28e20T^{2} \) |
| 5 | \( 1 + 8.77e14T + 1.13e30T^{2} \) |
| 11 | \( 1 - 2.59e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 1.25e24T + 7.93e47T^{2} \) |
| 17 | \( 1 - 1.55e26T + 8.11e52T^{2} \) |
| 19 | \( 1 + 3.27e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 9.13e28T + 3.58e58T^{2} \) |
| 29 | \( 1 - 4.60e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 9.90e31T + 1.34e64T^{2} \) |
| 37 | \( 1 + 1.17e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 3.11e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 1.85e35T + 1.73e70T^{2} \) |
| 47 | \( 1 - 1.01e36T + 7.94e71T^{2} \) |
| 53 | \( 1 - 8.71e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 1.22e38T + 1.40e76T^{2} \) |
| 61 | \( 1 + 2.33e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 4.76e38T + 3.32e78T^{2} \) |
| 71 | \( 1 + 4.50e39T + 4.01e79T^{2} \) |
| 73 | \( 1 - 1.43e40T + 1.32e80T^{2} \) |
| 79 | \( 1 + 1.20e41T + 3.96e81T^{2} \) |
| 83 | \( 1 + 6.29e40T + 3.31e82T^{2} \) |
| 89 | \( 1 + 5.31e41T + 6.66e83T^{2} \) |
| 97 | \( 1 + 2.11e42T + 2.69e85T^{2} \) |
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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
−11.45742939447761334387055954514, −10.42080619635626255491799810173, −8.894719289144715692626500548734, −8.205772732309953665362711640115, −6.71549006246719528397667605751, −5.88104301941881634680144999170, −4.08062247379992267029222191952, −3.12616234156629445995470037707, −1.53388693768990336573572175316, −0.44019237250485438074065282989,
0.44019237250485438074065282989, 1.53388693768990336573572175316, 3.12616234156629445995470037707, 4.08062247379992267029222191952, 5.88104301941881634680144999170, 6.71549006246719528397667605751, 8.205772732309953665362711640115, 8.894719289144715692626500548734, 10.42080619635626255491799810173, 11.45742939447761334387055954514
