| L(s) = 1 | + 25·5-s + 85.5·7-s − 99.8·11-s − 546.·13-s − 424.·17-s + 2.12e3·19-s − 3.45e3·23-s + 625·25-s + 6.87e3·29-s + 2.98e3·31-s + 2.13e3·35-s − 1.42e4·37-s + 423.·41-s − 3.56e3·43-s − 928.·47-s − 9.48e3·49-s − 3.32e4·53-s − 2.49e3·55-s − 1.78e4·59-s + 3.02e4·61-s − 1.36e4·65-s + 2.07e4·67-s + 1.92e4·71-s − 8.31e4·73-s − 8.54e3·77-s + 4.25e4·79-s − 9.35e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.659·7-s − 0.248·11-s − 0.896·13-s − 0.356·17-s + 1.34·19-s − 1.35·23-s + 0.200·25-s + 1.51·29-s + 0.558·31-s + 0.295·35-s − 1.70·37-s + 0.0393·41-s − 0.294·43-s − 0.0613·47-s − 0.564·49-s − 1.62·53-s − 0.111·55-s − 0.666·59-s + 1.04·61-s − 0.400·65-s + 0.565·67-s + 0.452·71-s − 1.82·73-s − 0.164·77-s + 0.766·79-s − 1.49·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 7 | \( 1 - 85.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 99.8T + 1.61e5T^{2} \) |
| 13 | \( 1 + 546.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 424.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 423.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 928.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.92e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.67e4T + 8.58e9T^{2} \) |
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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
−8.655839943147820546704809528965, −7.949486023037907980888340180392, −7.11530545904212682959362853925, −6.17990760770706044937820075907, −5.17204782195912322414206774059, −4.61090072866980858870119783762, −3.27342826975142905638178298159, −2.26735498587136408784369542264, −1.31592673365378363861948691499, 0,
1.31592673365378363861948691499, 2.26735498587136408784369542264, 3.27342826975142905638178298159, 4.61090072866980858870119783762, 5.17204782195912322414206774059, 6.17990760770706044937820075907, 7.11530545904212682959362853925, 7.949486023037907980888340180392, 8.655839943147820546704809528965
