L(s) = 1 | − 3-s − 3·5-s + 3·7-s + 9-s + 11-s − 2·13-s + 3·15-s − 5·17-s − 19-s − 3·21-s + 4·23-s + 4·25-s − 27-s − 6·29-s + 2·31-s − 33-s − 9·35-s + 8·37-s + 2·39-s − 8·41-s − 13·43-s − 3·45-s − 13·47-s + 2·49-s + 5·51-s − 6·53-s − 3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.774·15-s − 1.21·17-s − 0.229·19-s − 0.654·21-s + 0.834·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.174·33-s − 1.52·35-s + 1.31·37-s + 0.320·39-s − 1.24·41-s − 1.98·43-s − 0.447·45-s − 1.89·47-s + 2/7·49-s + 0.700·51-s − 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.711262015557123060475191370250, −8.640327447175563877730005802589, −7.955817008676270723012291209611, −7.20317748197754994750589075134, −6.33728946271749100225855748841, −4.85416520661419855074390038735, −4.60091553848511709420213798982, −3.37440380659751157658431848795, −1.72460303214518033154730596246, 0,
1.72460303214518033154730596246, 3.37440380659751157658431848795, 4.60091553848511709420213798982, 4.85416520661419855074390038735, 6.33728946271749100225855748841, 7.20317748197754994750589075134, 7.955817008676270723012291209611, 8.640327447175563877730005802589, 9.711262015557123060475191370250