| L(s) = 1 | − 1.70·2-s + 3·3-s − 5.10·4-s − 5·5-s − 5.10·6-s + 7·7-s + 22.2·8-s + 9·9-s + 8.50·10-s + 37.4·11-s − 15.3·12-s + 29.0·13-s − 11.9·14-s − 15·15-s + 2.89·16-s + 58.4·17-s − 15.3·18-s − 54.5·19-s + 25.5·20-s + 21·21-s − 63.6·22-s + 161.·23-s + 66.8·24-s + 25·25-s − 49.3·26-s + 27·27-s − 35.7·28-s + ⋯ |
| L(s) = 1 | − 0.601·2-s + 0.577·3-s − 0.638·4-s − 0.447·5-s − 0.347·6-s + 0.377·7-s + 0.985·8-s + 0.333·9-s + 0.269·10-s + 1.02·11-s − 0.368·12-s + 0.619·13-s − 0.227·14-s − 0.258·15-s + 0.0452·16-s + 0.833·17-s − 0.200·18-s − 0.659·19-s + 0.285·20-s + 0.218·21-s − 0.616·22-s + 1.46·23-s + 0.568·24-s + 0.200·25-s − 0.372·26-s + 0.192·27-s − 0.241·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.258616063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258616063\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 1.70T + 8T^{2} \) |
| 11 | \( 1 - 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 518.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 542.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 14.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−13.42975125679078246543569448949, −12.25656411438848290036343603269, −10.99283288889158405781600786128, −9.849320605974813936721103228053, −8.760606868012696065345540490324, −8.165518684722344472648947709944, −6.81635547436989365399984430237, −4.83096030102797826311073591617, −3.57515772035303027225893561748, −1.20229461864017516428374702022,
1.20229461864017516428374702022, 3.57515772035303027225893561748, 4.83096030102797826311073591617, 6.81635547436989365399984430237, 8.165518684722344472648947709944, 8.760606868012696065345540490324, 9.849320605974813936721103228053, 10.99283288889158405781600786128, 12.25656411438848290036343603269, 13.42975125679078246543569448949
