| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 11-s − 13-s − 15-s − 5·17-s + 2·19-s + 21-s + 7·23-s + 25-s − 27-s + 4·31-s + 33-s − 35-s − 7·37-s + 39-s − 11·41-s + 6·43-s + 45-s − 6·49-s + 5·51-s + 11·53-s − 55-s − 2·57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s − 1.21·17-s + 0.458·19-s + 0.218·21-s + 1.45·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s + 0.174·33-s − 0.169·35-s − 1.15·37-s + 0.160·39-s − 1.71·41-s + 0.914·43-s + 0.149·45-s − 6/7·49-s + 0.700·51-s + 1.51·53-s − 0.134·55-s − 0.264·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−7.52497049284418959018797916221, −6.76932515933827549795396792826, −6.48002794249954885136321718435, −5.37480164731559533044183587890, −5.05706220517884585286969524263, −4.13233725858994189975801585819, −3.12826551564954722904304742529, −2.32797933687062497279664947022, −1.23471748290718565818333854990, 0,
1.23471748290718565818333854990, 2.32797933687062497279664947022, 3.12826551564954722904304742529, 4.13233725858994189975801585819, 5.05706220517884585286969524263, 5.37480164731559533044183587890, 6.48002794249954885136321718435, 6.76932515933827549795396792826, 7.52497049284418959018797916221
