| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 6·13-s + 16-s + 2·20-s − 8·23-s − 25-s − 6·26-s − 6·29-s − 8·31-s − 32-s − 10·37-s − 2·40-s + 6·41-s + 12·43-s + 8·46-s + 50-s + 6·52-s + 10·53-s + 6·58-s − 8·59-s + 6·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 1.66·13-s + 1/4·16-s + 0.447·20-s − 1.66·23-s − 1/5·25-s − 1.17·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.64·37-s − 0.316·40-s + 0.937·41-s + 1.82·43-s + 1.17·46-s + 0.141·50-s + 0.832·52-s + 1.37·53-s + 0.787·58-s − 1.04·59-s + 0.768·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.07805492960575, −12.57011003530401, −12.14380768217627, −11.56361438705633, −11.04565792672035, −10.66576785866216, −10.40877908495603, −9.674051555801928, −9.354562363969502, −8.986628730582904, −8.459393318571215, −7.993541343898705, −7.471708595425051, −6.998847023300678, −6.343649363815251, −5.871366899575049, −5.718631016852612, −5.135050818560110, −4.100490901921939, −3.796356708584116, −3.350742792870399, −2.278465444952121, −2.104027790590142, −1.497354069133093, −0.8384251260041477, 0,
0.8384251260041477, 1.497354069133093, 2.104027790590142, 2.278465444952121, 3.350742792870399, 3.796356708584116, 4.100490901921939, 5.135050818560110, 5.718631016852612, 5.871366899575049, 6.343649363815251, 6.998847023300678, 7.471708595425051, 7.993541343898705, 8.459393318571215, 8.986628730582904, 9.354562363969502, 9.674051555801928, 10.40877908495603, 10.66576785866216, 11.04565792672035, 11.56361438705633, 12.14380768217627, 12.57011003530401, 13.07805492960575
