| L(s) = 1 | − 3·3-s + 4·5-s + 6·9-s − 11-s + 3·13-s − 12·15-s + 17-s − 2·19-s + 4·23-s + 11·25-s − 9·27-s − 8·31-s + 3·33-s − 8·37-s − 9·39-s − 10·43-s + 24·45-s + 10·47-s − 3·51-s − 3·53-s − 4·55-s + 6·57-s − 14·59-s − 8·61-s + 12·65-s + 10·67-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 2·9-s − 0.301·11-s + 0.832·13-s − 3.09·15-s + 0.242·17-s − 0.458·19-s + 0.834·23-s + 11/5·25-s − 1.73·27-s − 1.43·31-s + 0.522·33-s − 1.31·37-s − 1.44·39-s − 1.52·43-s + 3.57·45-s + 1.45·47-s − 0.420·51-s − 0.412·53-s − 0.539·55-s + 0.794·57-s − 1.82·59-s − 1.02·61-s + 1.48·65-s + 1.22·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.71630994327367, −13.93276124180869, −13.64167046311611, −13.11722203649009, −12.53315304170871, −12.36355333054176, −11.53720359389839, −10.83614949103060, −10.71672095353002, −10.34256967810793, −9.565293569664489, −9.235147353712027, −8.620791910265423, −7.749663744744249, −6.966561903888438, −6.577067707929489, −6.136528174987923, −5.630866376329901, −5.135767730584462, −4.899055031346820, −3.893656902047816, −3.156110528214242, −2.152173187841986, −1.592894522237906, −1.016427102458997, 0,
1.016427102458997, 1.592894522237906, 2.152173187841986, 3.156110528214242, 3.893656902047816, 4.899055031346820, 5.135767730584462, 5.630866376329901, 6.136528174987923, 6.577067707929489, 6.966561903888438, 7.749663744744249, 8.620791910265423, 9.235147353712027, 9.565293569664489, 10.34256967810793, 10.71672095353002, 10.83614949103060, 11.53720359389839, 12.36355333054176, 12.53315304170871, 13.11722203649009, 13.64167046311611, 13.93276124180869, 14.71630994327367
