| L(s) = 1 | + 4·2-s + 16·4-s − 103.·5-s + 64·8-s − 413.·10-s − 240.·11-s + 805.·13-s + 256·16-s + 1.29e3·17-s − 275.·19-s − 1.65e3·20-s − 960.·22-s − 3.79e3·23-s + 7.58e3·25-s + 3.22e3·26-s − 1.22e3·29-s + 5.62e3·31-s + 1.02e3·32-s + 5.17e3·34-s − 9.07e3·37-s − 1.10e3·38-s − 6.62e3·40-s + 1.82e4·41-s − 1.17e4·43-s − 3.84e3·44-s − 1.51e4·46-s + 2.30e4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.85·5-s + 0.353·8-s − 1.30·10-s − 0.598·11-s + 1.32·13-s + 0.250·16-s + 1.08·17-s − 0.175·19-s − 0.925·20-s − 0.423·22-s − 1.49·23-s + 2.42·25-s + 0.934·26-s − 0.271·29-s + 1.05·31-s + 0.176·32-s + 0.767·34-s − 1.09·37-s − 0.123·38-s − 0.654·40-s + 1.69·41-s − 0.965·43-s − 0.299·44-s − 1.05·46-s + 1.52·47-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 | \( 1 + 103.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 240.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 805.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.29e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 275.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.76e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.96e4T + 8.58e9T^{2} \) |
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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
−8.590615522744635089646514918710, −8.016107556012015818949064595035, −7.38961933418289585681318955410, −6.31815670830155299166156541536, −5.36774037980630252284283547740, −4.20775747082601207721577958367, −3.74490234916446448722665794607, −2.81355463206960669057603142382, −1.19102157271984553626278081789, 0,
1.19102157271984553626278081789, 2.81355463206960669057603142382, 3.74490234916446448722665794607, 4.20775747082601207721577958367, 5.36774037980630252284283547740, 6.31815670830155299166156541536, 7.38961933418289585681318955410, 8.016107556012015818949064595035, 8.590615522744635089646514918710
