| L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 11-s + 12-s + 2·13-s + 4·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 4·20-s − 22-s + 8·23-s − 24-s + 11·25-s − 2·26-s + 27-s + 2·29-s − 4·30-s − 2·31-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 + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.894·20-s − 0.213·22-s + 1.66·23-s − 0.204·24-s + 11/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.730·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.863115931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.863115931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.608718948017794829733912841514, −7.908066673185964494948274268454, −6.74333404085984723786614245803, −6.60589808110925983728983232275, −5.46172575196020262329273859280, −4.98947488697477787376362606595, −3.41920990896705159398090407919, −2.83750008917948826612427422392, −1.73178686189749874158490306737, −1.20452717923265751181000511481,
1.20452717923265751181000511481, 1.73178686189749874158490306737, 2.83750008917948826612427422392, 3.41920990896705159398090407919, 4.98947488697477787376362606595, 5.46172575196020262329273859280, 6.60589808110925983728983232275, 6.74333404085984723786614245803, 7.908066673185964494948274268454, 8.608718948017794829733912841514
