| L(s) = 1 | − 4·2-s + 16·4-s + 22.4·5-s − 64·8-s − 89.9·10-s − 340.·11-s + 728.·13-s + 256·16-s + 809.·17-s − 1.02e3·19-s + 359.·20-s + 1.36e3·22-s − 1.42e3·23-s − 2.61e3·25-s − 2.91e3·26-s − 5.21e3·29-s + 7.03e3·31-s − 1.02e3·32-s − 3.23e3·34-s + 1.27e4·37-s + 4.10e3·38-s − 1.43e3·40-s + 1.17e3·41-s + 3.66e3·43-s − 5.44e3·44-s + 5.68e3·46-s + 9.31e3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.402·5-s − 0.353·8-s − 0.284·10-s − 0.848·11-s + 1.19·13-s + 0.250·16-s + 0.679·17-s − 0.652·19-s + 0.201·20-s + 0.599·22-s − 0.560·23-s − 0.838·25-s − 0.845·26-s − 1.15·29-s + 1.31·31-s − 0.176·32-s − 0.480·34-s + 1.53·37-s + 0.461·38-s − 0.142·40-s + 0.109·41-s + 0.302·43-s − 0.424·44-s + 0.396·46-s + 0.614·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 22.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 340.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 728.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 809.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.27e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.31e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.61e5T + 8.58e9T^{2} \) |
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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
−9.028836540491247939619092079475, −8.055112854525507532072218935418, −7.60495344669563393371368526228, −6.18638740186039916205990629628, −5.88171352231280897524639204479, −4.49425724696113673905834650185, −3.30090166998639905152420973040, −2.21931967658440275607589337776, −1.21058453152264209795029917407, 0,
1.21058453152264209795029917407, 2.21931967658440275607589337776, 3.30090166998639905152420973040, 4.49425724696113673905834650185, 5.88171352231280897524639204479, 6.18638740186039916205990629628, 7.60495344669563393371368526228, 8.055112854525507532072218935418, 9.028836540491247939619092079475
