| L(s) = 1 | − 2-s − 3.37·3-s + 4-s + 5-s + 3.37·6-s + 3.37·7-s − 8-s + 8.37·9-s − 10-s − 11-s − 3.37·12-s + 2·13-s − 3.37·14-s − 3.37·15-s + 16-s + 1.37·17-s − 8.37·18-s + 0.627·19-s + 20-s − 11.3·21-s + 22-s + 2.74·23-s + 3.37·24-s + 25-s − 2·26-s − 18.1·27-s + 3.37·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.94·3-s + 0.5·4-s + 0.447·5-s + 1.37·6-s + 1.27·7-s − 0.353·8-s + 2.79·9-s − 0.316·10-s − 0.301·11-s − 0.973·12-s + 0.554·13-s − 0.901·14-s − 0.870·15-s + 0.250·16-s + 0.332·17-s − 1.97·18-s + 0.144·19-s + 0.223·20-s − 2.48·21-s + 0.213·22-s + 0.572·23-s + 0.688·24-s + 0.200·25-s − 0.392·26-s − 3.48·27-s + 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5403746435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5403746435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 9.37T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−13.41226572909749650300357866783, −12.18774721888558928121260895031, −11.34650080369151156579970477622, −10.72981264238965181959807392919, −9.768560529356435508292557762357, −8.104515689787382320128457509138, −6.84241733138923701159262011736, −5.72823185862067133238861064161, −4.73978172601008634193291616144, −1.34474408898394847193415655073,
1.34474408898394847193415655073, 4.73978172601008634193291616144, 5.72823185862067133238861064161, 6.84241733138923701159262011736, 8.104515689787382320128457509138, 9.768560529356435508292557762357, 10.72981264238965181959807392919, 11.34650080369151156579970477622, 12.18774721888558928121260895031, 13.41226572909749650300357866783
