| L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 9-s − 11-s − 4·13-s + 4·15-s − 2·21-s − 8·23-s − 25-s − 4·27-s − 6·29-s + 2·31-s − 2·33-s − 2·35-s − 10·37-s − 8·39-s − 4·43-s + 2·45-s − 2·47-s + 49-s − 10·53-s − 2·55-s + 10·59-s + 8·61-s − 63-s − 8·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 1.03·15-s − 0.436·21-s − 1.66·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.338·35-s − 1.64·37-s − 1.28·39-s − 0.609·43-s + 0.298·45-s − 0.291·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s + 1.30·59-s + 1.02·61-s − 0.125·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.975077527341635463849211087915, −7.33276918757842112321709021129, −6.48568106079950354332828940134, −5.69489782779642538238959922266, −5.03205189028723913882301778535, −3.90930896043299856799196866918, −3.23291466948871714952759127167, −2.23564297261560483384130965590, −1.92339283171851699345019773580, 0,
1.92339283171851699345019773580, 2.23564297261560483384130965590, 3.23291466948871714952759127167, 3.90930896043299856799196866918, 5.03205189028723913882301778535, 5.69489782779642538238959922266, 6.48568106079950354332828940134, 7.33276918757842112321709021129, 7.975077527341635463849211087915
