| L(s) = 1 | − 3-s − 5-s − 2.24·7-s + 9-s + 1.86·11-s + 4.82·13-s + 15-s − 6.69·17-s − 5.44·19-s + 2.24·21-s + 5.48·23-s + 25-s − 27-s + 3.91·29-s + 4.93·31-s − 1.86·33-s + 2.24·35-s − 6.15·37-s − 4.82·39-s − 9.11·41-s + 2.66·43-s − 45-s + 7.75·47-s − 1.96·49-s + 6.69·51-s + 13.4·53-s − 1.86·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.847·7-s + 0.333·9-s + 0.563·11-s + 1.33·13-s + 0.258·15-s − 1.62·17-s − 1.24·19-s + 0.489·21-s + 1.14·23-s + 0.200·25-s − 0.192·27-s + 0.726·29-s + 0.886·31-s − 0.325·33-s + 0.379·35-s − 1.01·37-s − 0.772·39-s − 1.42·41-s + 0.406·43-s − 0.149·45-s + 1.13·47-s − 0.281·49-s + 0.936·51-s + 1.84·53-s − 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 + 6.15T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 - 2.66T + 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 + 5.01T + 79T^{2} \) |
| 83 | \( 1 - 0.956T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 + 2.38T + 97T^{2} \) |
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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
−8.287421235498489834799551336651, −6.96933437148759656427738220156, −6.61763220262517401263031533456, −6.12688404433770497962127108656, −5.01026152142812075271548104674, −4.20723533261937447349659098189, −3.59766312145973420592196322780, −2.51318827536890223534225440661, −1.21734276967161242142852155127, 0,
1.21734276967161242142852155127, 2.51318827536890223534225440661, 3.59766312145973420592196322780, 4.20723533261937447349659098189, 5.01026152142812075271548104674, 6.12688404433770497962127108656, 6.61763220262517401263031533456, 6.96933437148759656427738220156, 8.287421235498489834799551336651
