| L(s) = 1 | + 14.5·2-s + 81·3-s − 300.·4-s − 625·5-s + 1.17e3·6-s − 2.40e3·7-s − 1.18e4·8-s + 6.56e3·9-s − 9.09e3·10-s − 4.12e4·11-s − 2.43e4·12-s + 7.59e4·13-s − 3.49e4·14-s − 5.06e4·15-s − 1.80e4·16-s + 3.23e5·17-s + 9.54e4·18-s − 3.37e5·19-s + 1.87e5·20-s − 1.94e5·21-s − 5.99e5·22-s + 1.37e6·23-s − 9.57e5·24-s + 3.90e5·25-s + 1.10e6·26-s + 5.31e5·27-s + 7.21e5·28-s + ⋯ |
| L(s) = 1 | + 0.642·2-s + 0.577·3-s − 0.586·4-s − 0.447·5-s + 0.371·6-s − 0.377·7-s − 1.01·8-s + 0.333·9-s − 0.287·10-s − 0.849·11-s − 0.338·12-s + 0.737·13-s − 0.242·14-s − 0.258·15-s − 0.0687·16-s + 0.938·17-s + 0.214·18-s − 0.594·19-s + 0.262·20-s − 0.218·21-s − 0.545·22-s + 1.02·23-s − 0.588·24-s + 0.200·25-s + 0.474·26-s + 0.192·27-s + 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.343883970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.343883970\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 14.5T + 512T^{2} \) |
| 11 | \( 1 + 4.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.59e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.23e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.37e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.37e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.27e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.36e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.19e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.42e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.01e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.33e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.27e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.03e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.65e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.27e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.13e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.50e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.33e8T + 7.60e17T^{2} \) |
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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
−12.40089517141406650666935966643, −10.95466149215134014122233297591, −9.711944946672934244152086787062, −8.671822273084131580235478676869, −7.71925233356516421632786096976, −6.16649207487421274419501113599, −4.88899681022675589491323719984, −3.73237178779847493527608566038, −2.79448417812480357008461684529, −0.75026291510493518582867650787,
0.75026291510493518582867650787, 2.79448417812480357008461684529, 3.73237178779847493527608566038, 4.88899681022675589491323719984, 6.16649207487421274419501113599, 7.71925233356516421632786096976, 8.671822273084131580235478676869, 9.711944946672934244152086787062, 10.95466149215134014122233297591, 12.40089517141406650666935966643
