| L(s) = 1 | + 6·3-s + 5·5-s − 34·7-s + 9·9-s − 16·11-s − 58·13-s + 30·15-s − 70·17-s − 4·19-s − 204·21-s − 134·23-s + 25·25-s − 108·27-s + 242·29-s + 100·31-s − 96·33-s − 170·35-s + 438·37-s − 348·39-s − 138·41-s − 178·43-s + 45·45-s + 22·47-s + 813·49-s − 420·51-s − 162·53-s − 80·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.83·7-s + 1/3·9-s − 0.438·11-s − 1.23·13-s + 0.516·15-s − 0.998·17-s − 0.0482·19-s − 2.11·21-s − 1.21·23-s + 1/5·25-s − 0.769·27-s + 1.54·29-s + 0.579·31-s − 0.506·33-s − 0.821·35-s + 1.94·37-s − 1.42·39-s − 0.525·41-s − 0.631·43-s + 0.149·45-s + 0.0682·47-s + 2.37·49-s − 1.15·51-s − 0.419·53-s − 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 134 T + p^{3} T^{2} \) |
| 29 | \( 1 - 242 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 438 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 22 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 268 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 422 T + p^{3} T^{2} \) |
| 71 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 73 | \( 1 - 306 T + p^{3} T^{2} \) |
| 79 | \( 1 + 456 T + p^{3} T^{2} \) |
| 83 | \( 1 + 434 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1378 T + p^{3} T^{2} \) |
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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
−10.23014463176491978369381799521, −9.781885963717609691940987832204, −8.992234361736871636430047539745, −7.984125124870024250470059182788, −6.84662389215761778858782885596, −5.98525973062276473797780306416, −4.38194918512963248962026338711, −2.99401994143364764477955915129, −2.42446982631790162680635176093, 0,
2.42446982631790162680635176093, 2.99401994143364764477955915129, 4.38194918512963248962026338711, 5.98525973062276473797780306416, 6.84662389215761778858782885596, 7.984125124870024250470059182788, 8.992234361736871636430047539745, 9.781885963717609691940987832204, 10.23014463176491978369381799521
