| L(s) = 1 | + 3-s − 2·9-s − 3·11-s + 3·13-s + 17-s − 6·19-s + 6·23-s − 5·27-s + 5·29-s − 3·33-s + 2·37-s + 3·39-s − 41-s − 8·43-s + 13·47-s + 51-s + 4·53-s − 6·57-s + 2·61-s − 2·67-s + 6·69-s + 8·71-s − 10·73-s + 11·79-s + 81-s + 5·87-s + 2·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.904·11-s + 0.832·13-s + 0.242·17-s − 1.37·19-s + 1.25·23-s − 0.962·27-s + 0.928·29-s − 0.522·33-s + 0.328·37-s + 0.480·39-s − 0.156·41-s − 1.21·43-s + 1.89·47-s + 0.140·51-s + 0.549·53-s − 0.794·57-s + 0.256·61-s − 0.244·67-s + 0.722·69-s + 0.949·71-s − 1.17·73-s + 1.23·79-s + 1/9·81-s + 0.536·87-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.25437963010968, −13.07779320908881, −12.31873589401042, −11.98321529358179, −11.34932403761163, −10.76368530018514, −10.64797673240959, −10.09722731971090, −9.300742538964176, −9.044189914467176, −8.456488461259643, −8.176922647055885, −7.782061874497572, −6.980507679282161, −6.633565794041164, −6.039693253391121, −5.437283474111923, −5.130235905467635, −4.323979469253952, −3.896550824070384, −3.226424437284656, −2.693258613572271, −2.377702398056851, −1.548193712460184, −0.8072012372637055, 0,
0.8072012372637055, 1.548193712460184, 2.377702398056851, 2.693258613572271, 3.226424437284656, 3.896550824070384, 4.323979469253952, 5.130235905467635, 5.437283474111923, 6.039693253391121, 6.633565794041164, 6.980507679282161, 7.782061874497572, 8.176922647055885, 8.456488461259643, 9.044189914467176, 9.300742538964176, 10.09722731971090, 10.64797673240959, 10.76368530018514, 11.34932403761163, 11.98321529358179, 12.31873589401042, 13.07779320908881, 13.25437963010968
