| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s − 2·11-s − 2·13-s + 15-s − 8·19-s + 2·21-s + 23-s + 25-s + 27-s − 10·29-s − 8·31-s − 2·33-s + 2·35-s + 8·37-s − 2·39-s − 6·41-s − 12·43-s + 45-s − 8·47-s − 3·49-s + 10·53-s − 2·55-s − 8·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.258·15-s − 1.83·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.348·33-s + 0.338·35-s + 1.31·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s − 1.16·47-s − 3/7·49-s + 1.37·53-s − 0.269·55-s − 1.05·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.81417203762081308908148277407, −7.24107948248560444101615145715, −6.41988324886864550344016879934, −5.52932896154542070844987186365, −4.89082478245225597485246334794, −4.11562648032117163784361171754, −3.20071799127292819293597680832, −2.13689586797702783654764076471, −1.75805536690367425869346834365, 0,
1.75805536690367425869346834365, 2.13689586797702783654764076471, 3.20071799127292819293597680832, 4.11562648032117163784361171754, 4.89082478245225597485246334794, 5.52932896154542070844987186365, 6.41988324886864550344016879934, 7.24107948248560444101615145715, 7.81417203762081308908148277407
