| L(s) = 1 | − 5·5-s + 22·7-s + 12·11-s + 38·13-s − 105·17-s + 157·19-s + 117·23-s + 25·25-s + 66·29-s + 25·31-s − 110·35-s + 314·37-s − 504·41-s − 380·43-s + 252·47-s + 141·49-s + 3·53-s − 60·55-s + 318·59-s + 293·61-s − 190·65-s + 322·67-s + 120·71-s + 44·73-s + 264·77-s − 917·79-s − 309·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.18·7-s + 0.328·11-s + 0.810·13-s − 1.49·17-s + 1.89·19-s + 1.06·23-s + 1/5·25-s + 0.422·29-s + 0.144·31-s − 0.531·35-s + 1.39·37-s − 1.91·41-s − 1.34·43-s + 0.782·47-s + 0.411·49-s + 0.00777·53-s − 0.147·55-s + 0.701·59-s + 0.614·61-s − 0.362·65-s + 0.587·67-s + 0.200·71-s + 0.0705·73-s + 0.390·77-s − 1.30·79-s − 0.408·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.746351556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.746351556\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 105 T + p^{3} T^{2} \) |
| 19 | \( 1 - 157 T + p^{3} T^{2} \) |
| 23 | \( 1 - 117 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 25 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 504 T + p^{3} T^{2} \) |
| 43 | \( 1 + 380 T + p^{3} T^{2} \) |
| 47 | \( 1 - 252 T + p^{3} T^{2} \) |
| 53 | \( 1 - 3 T + p^{3} T^{2} \) |
| 59 | \( 1 - 318 T + p^{3} T^{2} \) |
| 61 | \( 1 - 293 T + p^{3} T^{2} \) |
| 67 | \( 1 - 322 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 44 T + p^{3} T^{2} \) |
| 79 | \( 1 + 917 T + p^{3} T^{2} \) |
| 83 | \( 1 + 309 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1328 T + p^{3} T^{2} \) |
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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
−8.601591150513406660143105609660, −8.073984795241922239785229248594, −7.16796282077513354834222171152, −6.53246131072541104879264250647, −5.34185839777789793165124437247, −4.76298043839384812564481024238, −3.87223186896078592539082835511, −2.90333938493657302144748009078, −1.65853882783165728899243122584, −0.805261026022207019403575232192,
0.805261026022207019403575232192, 1.65853882783165728899243122584, 2.90333938493657302144748009078, 3.87223186896078592539082835511, 4.76298043839384812564481024238, 5.34185839777789793165124437247, 6.53246131072541104879264250647, 7.16796282077513354834222171152, 8.073984795241922239785229248594, 8.601591150513406660143105609660
