L(s) = 1 | − 3·3-s − 5-s + 7-s + 6·9-s − 3·11-s − 13-s + 3·15-s − 17-s + 4·19-s − 3·21-s − 4·23-s + 25-s − 9·27-s + 9·29-s − 6·31-s + 9·33-s − 35-s + 8·37-s + 3·39-s + 6·41-s − 8·43-s − 6·45-s + 7·47-s + 49-s + 3·51-s + 8·53-s + 3·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s − 0.904·11-s − 0.277·13-s + 0.774·15-s − 0.242·17-s + 0.917·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 1.07·31-s + 1.56·33-s − 0.169·35-s + 1.31·37-s + 0.480·39-s + 0.937·41-s − 1.21·43-s − 0.894·45-s + 1.02·47-s + 1/7·49-s + 0.420·51-s + 1.09·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.532599285516174297732711644887, −7.63092671882912581809859653130, −7.10658272673716100743817776181, −6.11844854812948446189006104628, −5.49185217337415225986235236520, −4.78517334215979002934325147021, −4.09342128574818473325708974171, −2.65615166372777141265130129061, −1.17540003477825799199948567871, 0,
1.17540003477825799199948567871, 2.65615166372777141265130129061, 4.09342128574818473325708974171, 4.78517334215979002934325147021, 5.49185217337415225986235236520, 6.11844854812948446189006104628, 7.10658272673716100743817776181, 7.63092671882912581809859653130, 8.532599285516174297732711644887