L(s) = 1 | − 2·2-s + 2·4-s − 3·11-s + 5·13-s − 4·16-s − 3·17-s − 2·19-s + 6·22-s − 23-s − 10·26-s + 6·29-s − 31-s + 8·32-s + 6·34-s + 4·38-s − 5·41-s + 43-s − 6·44-s + 2·46-s + 4·47-s − 7·49-s + 10·52-s − 5·53-s − 12·58-s + 12·59-s + 2·61-s + 2·62-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.904·11-s + 1.38·13-s − 16-s − 0.727·17-s − 0.458·19-s + 1.27·22-s − 0.208·23-s − 1.96·26-s + 1.11·29-s − 0.179·31-s + 1.41·32-s + 1.02·34-s + 0.648·38-s − 0.780·41-s + 0.152·43-s − 0.904·44-s + 0.294·46-s + 0.583·47-s − 49-s + 1.38·52-s − 0.686·53-s − 1.57·58-s + 1.56·59-s + 0.256·61-s + 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.60175460354004592995320915052, −6.72807204496435710554209117569, −6.33635364829565943158649495946, −5.35573397564159667548514874233, −4.56489225508159797223521187582, −3.75328871304018982360122163594, −2.71924510325995697326400282126, −1.92819919889618386525245427335, −1.02795689237807665559144453132, 0,
1.02795689237807665559144453132, 1.92819919889618386525245427335, 2.71924510325995697326400282126, 3.75328871304018982360122163594, 4.56489225508159797223521187582, 5.35573397564159667548514874233, 6.33635364829565943158649495946, 6.72807204496435710554209117569, 7.60175460354004592995320915052