| L(s) = 1 | − 3-s + 1.20·5-s − 7-s + 9-s + 2.02·11-s − 3.90·13-s − 1.20·15-s + 4.70·17-s − 4.02·19-s + 21-s − 23-s − 3.55·25-s − 27-s + 5.10·29-s − 5.10·31-s − 2.02·33-s − 1.20·35-s − 2.70·37-s + 3.90·39-s + 1.61·41-s − 6.55·43-s + 1.20·45-s + 7.13·47-s + 49-s − 4.70·51-s − 7.69·53-s + 2.43·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.538·5-s − 0.377·7-s + 0.333·9-s + 0.610·11-s − 1.08·13-s − 0.310·15-s + 1.14·17-s − 0.923·19-s + 0.218·21-s − 0.208·23-s − 0.710·25-s − 0.192·27-s + 0.948·29-s − 0.917·31-s − 0.352·33-s − 0.203·35-s − 0.444·37-s + 0.625·39-s + 0.252·41-s − 0.998·43-s + 0.179·45-s + 1.04·47-s + 0.142·49-s − 0.658·51-s − 1.05·53-s + 0.328·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.20T + 5T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 + 3.90T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 29 | \( 1 - 5.10T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 1.61T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 + 7.69T + 53T^{2} \) |
| 59 | \( 1 + 4.52T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 + 8.14T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 + 8.46T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 - 4.46T + 97T^{2} \) |
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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
−8.032042715203754192004815268147, −7.27212316478732168759727053509, −6.54806140404601815753244413314, −5.90981371509027856633873812808, −5.21903761948931423831867478226, −4.37509674961624981620949338964, −3.47483894118520078820508146602, −2.39560658683243561639163290476, −1.40953685899386292796649242799, 0,
1.40953685899386292796649242799, 2.39560658683243561639163290476, 3.47483894118520078820508146602, 4.37509674961624981620949338964, 5.21903761948931423831867478226, 5.90981371509027856633873812808, 6.54806140404601815753244413314, 7.27212316478732168759727053509, 8.032042715203754192004815268147
