| L(s) = 1 | + 4·5-s + 7-s + 3·11-s + 2·13-s − 4·17-s + 7·19-s + 23-s + 11·25-s + 10·29-s − 2·31-s + 4·35-s + 10·37-s − 7·41-s + 4·43-s − 3·47-s + 49-s − 9·53-s + 12·55-s + 9·59-s + 61-s + 8·65-s − 8·71-s + 4·73-s + 3·77-s + 14·79-s − 6·83-s − 16·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s − 0.970·17-s + 1.60·19-s + 0.208·23-s + 11/5·25-s + 1.85·29-s − 0.359·31-s + 0.676·35-s + 1.64·37-s − 1.09·41-s + 0.609·43-s − 0.437·47-s + 1/7·49-s − 1.23·53-s + 1.61·55-s + 1.17·59-s + 0.128·61-s + 0.992·65-s − 0.949·71-s + 0.468·73-s + 0.341·77-s + 1.57·79-s − 0.658·83-s − 1.73·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.885576548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.885576548\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.86310071737496, −13.37927607364226, −13.09048010132426, −12.40828484655767, −11.78550392638485, −11.41153530845025, −10.79393144528432, −10.36306282476186, −9.741380812447371, −9.343778609443861, −9.093856848000781, −8.352441489405846, −7.947527217040956, −6.949729762428437, −6.686437995222855, −6.215436034091431, −5.627464241475707, −5.153299708273924, −4.595982597724187, −3.966188101300949, −3.047790442313873, −2.666370925861931, −1.869310454448213, −1.322033961463087, −0.8435342658048839,
0.8435342658048839, 1.322033961463087, 1.869310454448213, 2.666370925861931, 3.047790442313873, 3.966188101300949, 4.595982597724187, 5.153299708273924, 5.627464241475707, 6.215436034091431, 6.686437995222855, 6.949729762428437, 7.947527217040956, 8.352441489405846, 9.093856848000781, 9.343778609443861, 9.741380812447371, 10.36306282476186, 10.79393144528432, 11.41153530845025, 11.78550392638485, 12.40828484655767, 13.09048010132426, 13.37927607364226, 13.86310071737496
