| L(s) = 1 | − 5-s − 4·7-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s + 6·37-s + 6·41-s − 8·43-s − 4·47-s + 9·49-s − 6·53-s + 4·55-s + 4·59-s − 2·61-s + 2·65-s + 8·67-s − 6·73-s + 16·77-s + 16·83-s + 2·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s + 0.248·65-s + 0.977·67-s − 0.702·73-s + 1.82·77-s + 1.75·83-s + 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.92459683746191757592598370055, −9.969026297182816734374055781991, −9.341400856476565034335427908941, −8.048390911320260435272830158835, −7.21857810559962191471917843249, −6.18178055218054403257438482099, −5.05802427658335260581937078682, −3.65788673520365604494122080438, −2.61772465980045468774058435809, 0,
2.61772465980045468774058435809, 3.65788673520365604494122080438, 5.05802427658335260581937078682, 6.18178055218054403257438482099, 7.21857810559962191471917843249, 8.048390911320260435272830158835, 9.341400856476565034335427908941, 9.969026297182816734374055781991, 10.92459683746191757592598370055
