| L(s) = 1 | + 3.02·2-s − 9.11·3-s + 1.15·4-s + 18.5·5-s − 27.5·6-s − 7·7-s − 20.7·8-s + 56.1·9-s + 56.2·10-s − 33.2·11-s − 10.5·12-s − 64.4·13-s − 21.1·14-s − 169.·15-s − 71.9·16-s − 52.2·17-s + 169.·18-s − 88.3·19-s + 21.4·20-s + 63.8·21-s − 100.·22-s + 23·23-s + 188.·24-s + 220.·25-s − 194.·26-s − 265.·27-s − 8.09·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 1.75·3-s + 0.144·4-s + 1.66·5-s − 1.87·6-s − 0.377·7-s − 0.915·8-s + 2.07·9-s + 1.77·10-s − 0.911·11-s − 0.253·12-s − 1.37·13-s − 0.404·14-s − 2.91·15-s − 1.12·16-s − 0.745·17-s + 2.22·18-s − 1.06·19-s + 0.240·20-s + 0.663·21-s − 0.975·22-s + 0.208·23-s + 1.60·24-s + 1.76·25-s − 1.47·26-s − 1.89·27-s − 0.0546·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 3.02T + 8T^{2} \) |
| 3 | \( 1 + 9.11T + 27T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 11 | \( 1 + 33.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 255.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 57.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 477.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 698.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 126.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 844.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 712.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 589.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 35.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 127.T + 9.12e5T^{2} \) |
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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
−12.36779701803946198772121460556, −10.95878906675814876029539210825, −10.17857757923759894550916070589, −9.250117571399743459187752888331, −6.90317378099379792089855723016, −6.06476488342851272780645922565, −5.33723112631325888815282099394, −4.61984609749904430370905339166, −2.35850675215200380312117848542, 0,
2.35850675215200380312117848542, 4.61984609749904430370905339166, 5.33723112631325888815282099394, 6.06476488342851272780645922565, 6.90317378099379792089855723016, 9.250117571399743459187752888331, 10.17857757923759894550916070589, 10.95878906675814876029539210825, 12.36779701803946198772121460556
