| L(s) = 1 | + 201.·3-s + 625·5-s − 2.30e3·7-s + 2.09e4·9-s − 2.17e4·11-s + 9.51e4·13-s + 1.25e5·15-s − 1.50e4·17-s + 1.12e6·19-s − 4.63e5·21-s + 2.00e6·23-s + 3.90e5·25-s + 2.53e5·27-s − 1.18e6·29-s + 1.92e6·31-s − 4.39e6·33-s − 1.43e6·35-s + 4.44e6·37-s + 1.91e7·39-s − 2.19e7·41-s + 2.53e7·43-s + 1.30e7·45-s − 3.64e7·47-s − 3.50e7·49-s − 3.03e6·51-s + 4.54e7·53-s − 1.36e7·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s + 0.447·5-s − 0.362·7-s + 1.06·9-s − 0.448·11-s + 0.924·13-s + 0.642·15-s − 0.0437·17-s + 1.97·19-s − 0.520·21-s + 1.49·23-s + 0.200·25-s + 0.0917·27-s − 0.311·29-s + 0.374·31-s − 0.644·33-s − 0.162·35-s + 0.390·37-s + 1.32·39-s − 1.21·41-s + 1.12·43-s + 0.475·45-s − 1.09·47-s − 0.868·49-s − 0.0628·51-s + 0.791·53-s − 0.200·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.894576081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.894576081\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| good | 3 | \( 1 - 201.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 2.30e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.17e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 9.51e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.50e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.12e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.92e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.44e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.19e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.53e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.64e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.79e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.35e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.76e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.42e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.25e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.55e9T + 7.60e17T^{2} \) |
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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
−13.03648815436483726393890411857, −11.38273765872708995286430923241, −9.952453989388518201095623764069, −9.141559742190333412587278347040, −8.144720079828455415709572703468, −6.96647734586273725142928649162, −5.35276953655564199319755521546, −3.55196807593875496463169321211, −2.68546669777463419980060802940, −1.19215639251239722226521382663,
1.19215639251239722226521382663, 2.68546669777463419980060802940, 3.55196807593875496463169321211, 5.35276953655564199319755521546, 6.96647734586273725142928649162, 8.144720079828455415709572703468, 9.141559742190333412587278347040, 9.952453989388518201095623764069, 11.38273765872708995286430923241, 13.03648815436483726393890411857
