| L(s) = 1 | − 1.56·3-s + 5-s + 0.438·7-s − 0.561·9-s + 4·11-s + 1.56·13-s − 1.56·15-s + 3.56·17-s + 19-s − 0.684·21-s + 6.68·23-s + 25-s + 5.56·27-s + 7.56·29-s + 3.12·31-s − 6.24·33-s + 0.438·35-s − 7.12·37-s − 2.43·39-s − 8.24·41-s + 2·43-s − 0.561·45-s − 5.12·47-s − 6.80·49-s − 5.56·51-s − 2.43·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.447·5-s + 0.165·7-s − 0.187·9-s + 1.20·11-s + 0.433·13-s − 0.403·15-s + 0.863·17-s + 0.229·19-s − 0.149·21-s + 1.39·23-s + 0.200·25-s + 1.07·27-s + 1.40·29-s + 0.560·31-s − 1.08·33-s + 0.0741·35-s − 1.17·37-s − 0.390·39-s − 1.28·41-s + 0.304·43-s − 0.0837·45-s − 0.747·47-s − 0.972·49-s − 0.778·51-s − 0.334·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.887819397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.887819397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 5.80T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−8.315507256576916796396892464541, −6.94535523318775817339876394312, −6.70848009831705645681524931654, −5.92937235678922846082190471563, −5.23427759648062209665367516142, −4.70884087838282243015241458265, −3.59370650440738886990955010702, −2.88509775242745812307144029991, −1.53306908255023097746391943587, −0.832187340095773646565405038223,
0.832187340095773646565405038223, 1.53306908255023097746391943587, 2.88509775242745812307144029991, 3.59370650440738886990955010702, 4.70884087838282243015241458265, 5.23427759648062209665367516142, 5.92937235678922846082190471563, 6.70848009831705645681524931654, 6.94535523318775817339876394312, 8.315507256576916796396892464541
