| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 8-s − 2·9-s + 4·10-s + 2·11-s + 12-s + 13-s + 4·15-s + 16-s − 3·17-s − 2·18-s + 19-s + 4·20-s + 2·22-s − 23-s + 24-s + 11·25-s + 26-s − 5·27-s − 5·29-s + 4·30-s + 8·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 1.26·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 1.03·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.229·19-s + 0.894·20-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s − 0.962·27-s − 0.928·29-s + 0.730·30-s + 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.452350475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.452350475\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.254038079040445697892143439151, −8.655339283587140731226502868667, −7.60323894240594246780554900498, −6.45399284641636243911038304495, −6.09315132100286987427486982952, −5.29912825884459151712444657250, −4.32145140400317761773094051986, −3.13458058385460483521784962114, −2.39336202780458240475893030948, −1.50913345329536755396223442365,
1.50913345329536755396223442365, 2.39336202780458240475893030948, 3.13458058385460483521784962114, 4.32145140400317761773094051986, 5.29912825884459151712444657250, 6.09315132100286987427486982952, 6.45399284641636243911038304495, 7.60323894240594246780554900498, 8.655339283587140731226502868667, 9.254038079040445697892143439151
