| L(s) = 1 | − 0.222·2-s − 1.95·4-s − 0.636·5-s + 1.72·7-s + 0.877·8-s + 0.141·10-s + 4.95·11-s + 2.50·13-s − 0.384·14-s + 3.70·16-s − 0.950·19-s + 1.24·20-s − 1.09·22-s − 2.12·23-s − 4.59·25-s − 0.556·26-s − 3.37·28-s + 9.58·29-s − 5.27·31-s − 2.57·32-s − 1.09·35-s − 8.48·37-s + 0.211·38-s − 0.558·40-s + 6.92·41-s + 7.15·43-s − 9.65·44-s + ⋯ |
| L(s) = 1 | − 0.157·2-s − 0.975·4-s − 0.284·5-s + 0.653·7-s + 0.310·8-s + 0.0447·10-s + 1.49·11-s + 0.695·13-s − 0.102·14-s + 0.926·16-s − 0.218·19-s + 0.277·20-s − 0.234·22-s − 0.442·23-s − 0.918·25-s − 0.109·26-s − 0.637·28-s + 1.78·29-s − 0.947·31-s − 0.455·32-s − 0.185·35-s − 1.39·37-s + 0.0342·38-s − 0.0883·40-s + 1.08·41-s + 1.09·43-s − 1.45·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484484629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484484629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.222T + 2T^{2} \) |
| 5 | \( 1 + 0.636T + 5T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 19 | \( 1 + 0.950T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 31 | \( 1 + 5.27T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 2.96T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 0.384T + 71T^{2} \) |
| 73 | \( 1 + 6.51T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 1.98T + 89T^{2} \) |
| 97 | \( 1 + 3.04T + 97T^{2} \) |
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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
−8.766178864770082516737361892409, −8.339834175034782842374560806571, −7.52441407798367566807955765790, −6.55471030891353130986603955890, −5.76392217517162998528855607608, −4.79255277283826474732865208162, −4.07063207948840865248369735412, −3.50496914994272462627180519163, −1.85083586631776684060365560938, −0.841696080020897582673071797337,
0.841696080020897582673071797337, 1.85083586631776684060365560938, 3.50496914994272462627180519163, 4.07063207948840865248369735412, 4.79255277283826474732865208162, 5.76392217517162998528855607608, 6.55471030891353130986603955890, 7.52441407798367566807955765790, 8.339834175034782842374560806571, 8.766178864770082516737361892409
