| L(s) = 1 | + 2·5-s + 2·7-s − 3·9-s − 11-s + 2·17-s − 8·19-s − 3·23-s − 25-s + 7·29-s + 7·31-s + 4·35-s + 37-s + 12·41-s − 8·43-s − 6·45-s − 3·49-s + 6·53-s − 2·55-s + 2·59-s + 7·61-s − 6·63-s − 6·67-s − 71-s + 11·73-s − 2·77-s − 8·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 9-s − 0.301·11-s + 0.485·17-s − 1.83·19-s − 0.625·23-s − 1/5·25-s + 1.29·29-s + 1.25·31-s + 0.676·35-s + 0.164·37-s + 1.87·41-s − 1.21·43-s − 0.894·45-s − 3/7·49-s + 0.824·53-s − 0.269·55-s + 0.260·59-s + 0.896·61-s − 0.755·63-s − 0.733·67-s − 0.118·71-s + 1.28·73-s − 0.227·77-s − 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582553997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582553997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.54013622087184, −13.17826132810694, −12.52049643676298, −12.10518060422156, −11.51821383801559, −11.19027543180866, −10.46981781572563, −10.23857704403717, −9.743444493878057, −9.033317373134854, −8.575631540110907, −8.122960373719819, −7.885416616169087, −6.971889140169486, −6.353715153160693, −6.047764943112848, −5.575422765513549, −4.906128612020470, −4.493635680616574, −3.848397235340960, −3.016791887556796, −2.381139595332902, −2.135042167957868, −1.275956443528768, −0.4877730261733812,
0.4877730261733812, 1.275956443528768, 2.135042167957868, 2.381139595332902, 3.016791887556796, 3.848397235340960, 4.493635680616574, 4.906128612020470, 5.575422765513549, 6.047764943112848, 6.353715153160693, 6.971889140169486, 7.885416616169087, 8.122960373719819, 8.575631540110907, 9.033317373134854, 9.743444493878057, 10.23857704403717, 10.46981781572563, 11.19027543180866, 11.51821383801559, 12.10518060422156, 12.52049643676298, 13.17826132810694, 13.54013622087184
