| L(s) = 1 | − 2.38·2-s + 3.69·4-s + 2.92·5-s − 4.05·8-s − 6.97·10-s − 1.35·11-s − 1.46·13-s + 2.27·16-s + 3.31·17-s − 2.20·19-s + 10.7·20-s + 3.23·22-s + 2.62·23-s + 3.53·25-s + 3.49·26-s − 1.04·29-s − 3.27·31-s + 2.67·32-s − 7.90·34-s − 10.8·37-s + 5.26·38-s − 11.8·40-s − 1.80·41-s + 4.34·43-s − 5.00·44-s − 6.27·46-s − 3.97·47-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.84·4-s + 1.30·5-s − 1.43·8-s − 2.20·10-s − 0.408·11-s − 0.406·13-s + 0.568·16-s + 0.802·17-s − 0.506·19-s + 2.41·20-s + 0.688·22-s + 0.548·23-s + 0.706·25-s + 0.686·26-s − 0.193·29-s − 0.588·31-s + 0.472·32-s − 1.35·34-s − 1.78·37-s + 0.854·38-s − 1.87·40-s − 0.282·41-s + 0.662·43-s − 0.754·44-s − 0.924·46-s − 0.580·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9837407045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9837407045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 1.04T + 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 - 4.34T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 - 6.45T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 0.559T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.767T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 - 6.40T + 89T^{2} \) |
| 97 | \( 1 + 8.28T + 97T^{2} \) |
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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
−8.625632234392087722292997559517, −7.899044432855382211854578774725, −7.10270387707879992165229044546, −6.56899241495835547619349150221, −5.62671593093323631076888505037, −5.03722609268601410893504373082, −3.51647942334479234411907203401, −2.35455975387711705533139319182, −1.86983448886703689203408467588, −0.73209775417989378333412672562,
0.73209775417989378333412672562, 1.86983448886703689203408467588, 2.35455975387711705533139319182, 3.51647942334479234411907203401, 5.03722609268601410893504373082, 5.62671593093323631076888505037, 6.56899241495835547619349150221, 7.10270387707879992165229044546, 7.899044432855382211854578774725, 8.625632234392087722292997559517
