| L(s) = 1 | + 2-s + 1.14·3-s + 4-s + 0.853·5-s + 1.14·6-s − 7-s + 8-s − 1.68·9-s + 0.853·10-s + 2.68·11-s + 1.14·12-s + 4.68·13-s − 14-s + 0.978·15-s + 16-s − 1.83·17-s − 1.68·18-s − 4.97·19-s + 0.853·20-s − 1.14·21-s + 2.68·22-s − 23-s + 1.14·24-s − 4.27·25-s + 4.68·26-s − 5.37·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.661·3-s + 0.5·4-s + 0.381·5-s + 0.468·6-s − 0.377·7-s + 0.353·8-s − 0.561·9-s + 0.269·10-s + 0.809·11-s + 0.330·12-s + 1.29·13-s − 0.267·14-s + 0.252·15-s + 0.250·16-s − 0.444·17-s − 0.397·18-s − 1.14·19-s + 0.190·20-s − 0.250·21-s + 0.572·22-s − 0.208·23-s + 0.234·24-s − 0.854·25-s + 0.918·26-s − 1.03·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.401258632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401258632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 - 0.853T + 5T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 + 4.97T + 19T^{2} \) |
| 29 | \( 1 + 6.39T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 3.43T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 + 0.335T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 - 3.53T + 89T^{2} \) |
| 97 | \( 1 - 2.75T + 97T^{2} \) |
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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
−11.58009815667265073011536694185, −10.94671759377651771835195135639, −9.638631314485461693466762282886, −8.822836966363495951808179858298, −7.87888576634086669408227056210, −6.39949913358060342686353334909, −5.93588797656866733639137224124, −4.27109699633311311183452821209, −3.34038887732375351274960806521, −1.99384503715364443603141936325,
1.99384503715364443603141936325, 3.34038887732375351274960806521, 4.27109699633311311183452821209, 5.93588797656866733639137224124, 6.39949913358060342686353334909, 7.87888576634086669408227056210, 8.822836966363495951808179858298, 9.638631314485461693466762282886, 10.94671759377651771835195135639, 11.58009815667265073011536694185
