| L(s) = 1 | − 4·2-s + 16·4-s + 77.5·5-s − 64·8-s − 310.·10-s − 618.·11-s + 130.·13-s + 256·16-s − 263.·17-s + 91.9·19-s + 1.24e3·20-s + 2.47e3·22-s + 1.39e3·23-s + 2.89e3·25-s − 521.·26-s + 5.69e3·29-s − 6.95e3·31-s − 1.02e3·32-s + 1.05e3·34-s + 5.04e3·37-s − 367.·38-s − 4.96e3·40-s + 2.09e4·41-s − 2.13e4·43-s − 9.89e3·44-s − 5.56e3·46-s − 1.98e4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.38·5-s − 0.353·8-s − 0.981·10-s − 1.54·11-s + 0.213·13-s + 0.250·16-s − 0.221·17-s + 0.0584·19-s + 0.694·20-s + 1.08·22-s + 0.548·23-s + 0.926·25-s − 0.151·26-s + 1.25·29-s − 1.29·31-s − 0.176·32-s + 0.156·34-s + 0.606·37-s − 0.0413·38-s − 0.490·40-s + 1.95·41-s − 1.76·43-s − 0.770·44-s − 0.387·46-s − 1.31·47-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 | \( 1 - 77.5T + 3.12e3T^{2} \) |
| 11 | \( 1 + 618.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 130.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 263.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 91.9T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.98e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.75e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.98e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.38e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.37e5T + 8.58e9T^{2} \) |
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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
−9.110987749020565277903405329118, −8.220400334419701025294000155163, −7.37080987351475880579049186904, −6.35577450239813307539667345171, −5.63181366803601933671415933532, −4.77707166905828990550874416558, −3.04556915047309937487484153527, −2.28413930294448249988341959805, −1.30001818799649595666258386704, 0,
1.30001818799649595666258386704, 2.28413930294448249988341959805, 3.04556915047309937487484153527, 4.77707166905828990550874416558, 5.63181366803601933671415933532, 6.35577450239813307539667345171, 7.37080987351475880579049186904, 8.220400334419701025294000155163, 9.110987749020565277903405329118
