| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s + 13-s − 15-s − 5·17-s − 2·19-s − 21-s + 7·23-s + 25-s + 27-s + 4·31-s + 33-s + 35-s + 7·37-s + 39-s − 11·41-s − 6·43-s − 45-s − 6·49-s − 5·51-s − 11·53-s − 55-s − 2·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s − 1.21·17-s − 0.458·19-s − 0.218·21-s + 1.45·23-s + 1/5·25-s + 0.192·27-s + 0.718·31-s + 0.174·33-s + 0.169·35-s + 1.15·37-s + 0.160·39-s − 1.71·41-s − 0.914·43-s − 0.149·45-s − 6/7·49-s − 0.700·51-s − 1.51·53-s − 0.134·55-s − 0.264·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−16.60907365606805, −15.84363091223878, −15.49036294781189, −14.90617708520097, −14.50661624437114, −13.66473442803788, −13.08185342508891, −12.96160021675163, −11.96569019848428, −11.48593948707557, −10.86996530457729, −10.27730738701398, −9.497070602207338, −9.026167031832423, −8.428045189565318, −7.943175563465726, −7.009930190045331, −6.663942108642823, −6.002670832130190, −4.776227379306044, −4.564541106923443, −3.516962505542049, −3.080526023584434, −2.169383345146139, −1.227089757592571, 0,
1.227089757592571, 2.169383345146139, 3.080526023584434, 3.516962505542049, 4.564541106923443, 4.776227379306044, 6.002670832130190, 6.663942108642823, 7.009930190045331, 7.943175563465726, 8.428045189565318, 9.026167031832423, 9.497070602207338, 10.27730738701398, 10.86996530457729, 11.48593948707557, 11.96569019848428, 12.96160021675163, 13.08185342508891, 13.66473442803788, 14.50661624437114, 14.90617708520097, 15.49036294781189, 15.84363091223878, 16.60907365606805
