| L(s) = 1 | + 3·5-s − 7-s + 7·13-s + 2·17-s − 19-s − 6·23-s + 4·25-s − 7·29-s − 10·31-s − 3·35-s − 7·37-s − 6·41-s − 8·43-s + 13·47-s + 49-s + 10·53-s + 9·59-s − 6·61-s + 21·65-s + 67-s − 6·71-s + 13·73-s + 8·79-s − 18·83-s + 6·85-s − 2·89-s − 7·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 1.94·13-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 4/5·25-s − 1.29·29-s − 1.79·31-s − 0.507·35-s − 1.15·37-s − 0.937·41-s − 1.21·43-s + 1.89·47-s + 1/7·49-s + 1.37·53-s + 1.17·59-s − 0.768·61-s + 2.60·65-s + 0.122·67-s − 0.712·71-s + 1.52·73-s + 0.900·79-s − 1.97·83-s + 0.650·85-s − 0.211·89-s − 0.733·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.31527444946882, −13.96618666127589, −13.50342815043508, −13.21473386050587, −12.65746224853678, −12.08100624507239, −11.46260657085161, −10.82701897540711, −10.49294846391473, −9.934874101851261, −9.482140405874717, −8.822071532995847, −8.598252927894676, −7.857962105924956, −7.052131553312852, −6.688249125139011, −5.901390400769969, −5.652826809689400, −5.350234921773911, −4.153607987279462, −3.702632832029347, −3.239308011766513, −2.135990787147094, −1.839216353908668, −1.126046803516465, 0,
1.126046803516465, 1.839216353908668, 2.135990787147094, 3.239308011766513, 3.702632832029347, 4.153607987279462, 5.350234921773911, 5.652826809689400, 5.901390400769969, 6.688249125139011, 7.052131553312852, 7.857962105924956, 8.598252927894676, 8.822071532995847, 9.482140405874717, 9.934874101851261, 10.49294846391473, 10.82701897540711, 11.46260657085161, 12.08100624507239, 12.65746224853678, 13.21473386050587, 13.50342815043508, 13.96618666127589, 14.31527444946882
