| L(s) = 1 | + 5·3-s + 7·5-s − 2·9-s − 11·11-s − 52·13-s + 35·15-s − 46·17-s + 96·19-s + 27·23-s − 76·25-s − 145·27-s + 16·29-s + 293·31-s − 55·33-s − 29·37-s − 260·39-s + 472·41-s − 110·43-s − 14·45-s + 224·47-s − 230·51-s + 754·53-s − 77·55-s + 480·57-s − 825·59-s + 548·61-s − 364·65-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 0.626·5-s − 0.0740·9-s − 0.301·11-s − 1.10·13-s + 0.602·15-s − 0.656·17-s + 1.15·19-s + 0.244·23-s − 0.607·25-s − 1.03·27-s + 0.102·29-s + 1.69·31-s − 0.290·33-s − 0.128·37-s − 1.06·39-s + 1.79·41-s − 0.390·43-s − 0.0463·45-s + 0.695·47-s − 0.631·51-s + 1.95·53-s − 0.188·55-s + 1.11·57-s − 1.82·59-s + 1.15·61-s − 0.694·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.221920872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.221920872\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 96 T + p^{3} T^{2} \) |
| 23 | \( 1 - 27 T + p^{3} T^{2} \) |
| 29 | \( 1 - 16 T + p^{3} T^{2} \) |
| 31 | \( 1 - 293 T + p^{3} T^{2} \) |
| 37 | \( 1 + 29 T + p^{3} T^{2} \) |
| 41 | \( 1 - 472 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 - 224 T + p^{3} T^{2} \) |
| 53 | \( 1 - 754 T + p^{3} T^{2} \) |
| 59 | \( 1 + 825 T + p^{3} T^{2} \) |
| 61 | \( 1 - 548 T + p^{3} T^{2} \) |
| 67 | \( 1 + 123 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1001 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 79 | \( 1 - 526 T + p^{3} T^{2} \) |
| 83 | \( 1 - 158 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1217 T + p^{3} T^{2} \) |
| 97 | \( 1 - 263 T + p^{3} T^{2} \) |
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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
−8.794266427745288570650119341614, −7.930730449002280849453570115533, −7.39899213381896901552690483251, −6.39562919451500966656812411301, −5.51238494649100799433347544718, −4.72825591134328527969285361270, −3.64173091144746530279267835125, −2.60230011025103489382875352299, −2.22735929181966407734513557795, −0.75493905879002333244016796085,
0.75493905879002333244016796085, 2.22735929181966407734513557795, 2.60230011025103489382875352299, 3.64173091144746530279267835125, 4.72825591134328527969285361270, 5.51238494649100799433347544718, 6.39562919451500966656812411301, 7.39899213381896901552690483251, 7.930730449002280849453570115533, 8.794266427745288570650119341614
