| L(s) = 1 | − 2-s + 1.72·3-s + 4-s + 5-s − 1.72·6-s − 1.60·7-s − 8-s − 0.00863·9-s − 10-s − 1.19·11-s + 1.72·12-s + 4.83·13-s + 1.60·14-s + 1.72·15-s + 16-s − 1.36·17-s + 0.00863·18-s − 1.45·19-s + 20-s − 2.77·21-s + 1.19·22-s − 1.72·24-s + 25-s − 4.83·26-s − 5.20·27-s − 1.60·28-s + 1.38·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.998·3-s + 0.5·4-s + 0.447·5-s − 0.706·6-s − 0.605·7-s − 0.353·8-s − 0.00287·9-s − 0.316·10-s − 0.361·11-s + 0.499·12-s + 1.34·13-s + 0.428·14-s + 0.446·15-s + 0.250·16-s − 0.331·17-s + 0.00203·18-s − 0.332·19-s + 0.223·20-s − 0.605·21-s + 0.255·22-s − 0.353·24-s + 0.200·25-s − 0.948·26-s − 1.00·27-s − 0.302·28-s + 0.257·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984903816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984903816\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 17 | \( 1 + 1.36T + 17T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 0.671T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 5.77T + 53T^{2} \) |
| 59 | \( 1 - 9.76T + 59T^{2} \) |
| 61 | \( 1 + 1.13T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 + 1.40T + 71T^{2} \) |
| 73 | \( 1 - 2.01T + 73T^{2} \) |
| 79 | \( 1 - 9.25T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 - 1.01T + 89T^{2} \) |
| 97 | \( 1 + 5.04T + 97T^{2} \) |
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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
−8.232414667541902612324378063406, −7.80549091978392323558852577230, −6.80997812254822794993936267674, −6.17445651754428902449056398311, −5.56987787721986573957573211097, −4.26691354317380736472508307677, −3.44680771034148024595067011707, −2.69914824214301392071174587060, −2.01469617221471141466706919120, −0.797671159542545558455229197249,
0.797671159542545558455229197249, 2.01469617221471141466706919120, 2.69914824214301392071174587060, 3.44680771034148024595067011707, 4.26691354317380736472508307677, 5.56987787721986573957573211097, 6.17445651754428902449056398311, 6.80997812254822794993936267674, 7.80549091978392323558852577230, 8.232414667541902612324378063406
