| L(s) = 1 | − 2·4-s + 5-s − 7-s + 4·16-s + 3·17-s + 2·19-s − 2·20-s + 6·23-s − 4·25-s + 2·28-s − 29-s − 8·31-s − 35-s + 2·37-s + 2·41-s + 43-s + 3·47-s + 49-s + 4·53-s − 3·59-s + 4·61-s − 8·64-s − 7·67-s − 6·68-s + 2·71-s + 2·73-s − 4·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s + 16-s + 0.727·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s − 4/5·25-s + 0.377·28-s − 0.185·29-s − 1.43·31-s − 0.169·35-s + 0.328·37-s + 0.312·41-s + 0.152·43-s + 0.437·47-s + 1/7·49-s + 0.549·53-s − 0.390·59-s + 0.512·61-s − 64-s − 0.855·67-s − 0.727·68-s + 0.237·71-s + 0.234·73-s − 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.09191587244932, −12.86272299061095, −12.47895214036828, −11.78827096557052, −11.43044136543312, −10.71759797197585, −10.32808640922040, −9.868469665603888, −9.393175470961583, −9.007430189869462, −8.768620085785792, −7.836148324335340, −7.654961312259080, −7.110074633542056, −6.374806645593432, −5.906320765581436, −5.364420982264655, −5.125718788954090, −4.419632509133516, −3.769892983718794, −3.463005020154517, −2.809936465989325, −2.117945718983303, −1.334553401629509, −0.8011439841735829, 0,
0.8011439841735829, 1.334553401629509, 2.117945718983303, 2.809936465989325, 3.463005020154517, 3.769892983718794, 4.419632509133516, 5.125718788954090, 5.364420982264655, 5.906320765581436, 6.374806645593432, 7.110074633542056, 7.654961312259080, 7.836148324335340, 8.768620085785792, 9.007430189869462, 9.393175470961583, 9.868469665603888, 10.32808640922040, 10.71759797197585, 11.43044136543312, 11.78827096557052, 12.47895214036828, 12.86272299061095, 13.09191587244932
