| L(s) = 1 | + 3-s − 3·5-s + 7-s − 2·9-s + 2·11-s + 3·13-s − 3·15-s − 6·17-s − 4·19-s + 21-s − 2·23-s + 4·25-s − 5·27-s − 6·29-s + 10·31-s + 2·33-s − 3·35-s − 2·37-s + 3·39-s − 10·41-s − 4·43-s + 6·45-s − 7·47-s − 6·49-s − 6·51-s + 8·53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.832·13-s − 0.774·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s + 4/5·25-s − 0.962·27-s − 1.11·29-s + 1.79·31-s + 0.348·33-s − 0.507·35-s − 0.328·37-s + 0.480·39-s − 1.56·41-s − 0.609·43-s + 0.894·45-s − 1.02·47-s − 6/7·49-s − 0.840·51-s + 1.09·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−8.806695473952422658633000158062, −8.564424981103890785422457175495, −7.903372833693644985808766966158, −6.86805667484759342647357469603, −6.09922127840546731746771292105, −4.70019585546845455212568823793, −3.98638135363332626512942033326, −3.18285249010353951279255341401, −1.87107040460257588490750995471, 0,
1.87107040460257588490750995471, 3.18285249010353951279255341401, 3.98638135363332626512942033326, 4.70019585546845455212568823793, 6.09922127840546731746771292105, 6.86805667484759342647357469603, 7.903372833693644985808766966158, 8.564424981103890785422457175495, 8.806695473952422658633000158062
