L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s + 2·7-s + 3·8-s + 9-s − 10-s − 6·11-s + 2·12-s − 2·13-s − 2·14-s − 2·15-s − 16-s + 6·17-s − 18-s − 4·19-s − 20-s − 4·21-s + 6·22-s + 2·23-s − 6·24-s + 25-s + 2·26-s + 4·27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s − 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 1.27·22-s + 0.417·23-s − 1.22·24-s + 1/5·25-s + 0.392·26-s + 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 1601 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.64180625039070646647934243011, −6.96077607234823661197585781998, −5.81304454420980602866309778738, −5.36408462683488464840556406717, −4.99949483583419118726258580216, −4.20484602851035183485167770697, −2.91849881417646146763263909653, −1.96063183251360655991191717354, −0.909193834741079949871477253890, 0,
0.909193834741079949871477253890, 1.96063183251360655991191717354, 2.91849881417646146763263909653, 4.20484602851035183485167770697, 4.99949483583419118726258580216, 5.36408462683488464840556406717, 5.81304454420980602866309778738, 6.96077607234823661197585781998, 7.64180625039070646647934243011