L(s) = 1 | + 3-s − 2·4-s + 3·5-s − 4·7-s + 9-s − 3·11-s − 2·12-s − 13-s + 3·15-s + 4·16-s − 17-s − 19-s − 6·20-s − 4·21-s + 9·23-s + 4·25-s + 27-s + 8·28-s + 6·29-s + 2·31-s − 3·33-s − 12·35-s − 2·36-s − 4·37-s − 39-s − 3·41-s − 7·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s − 1.51·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 0.277·13-s + 0.774·15-s + 16-s − 0.242·17-s − 0.229·19-s − 1.34·20-s − 0.872·21-s + 1.87·23-s + 4/5·25-s + 0.192·27-s + 1.51·28-s + 1.11·29-s + 0.359·31-s − 0.522·33-s − 2.02·35-s − 1/3·36-s − 0.657·37-s − 0.160·39-s − 0.468·41-s − 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8600590161\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8600590161\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.42382989853517912004077694458, −14.11125275639621900792162393084, −13.19069815363258335174666499645, −12.84688978513418512171462235404, −10.23827402727964604848387184191, −9.642727340101070443548067847258, −8.618966621189813626434174913816, −6.66463575826783700250676263362, −5.14306131647545403565477997005, −3.00632648437536992753290178800,
3.00632648437536992753290178800, 5.14306131647545403565477997005, 6.66463575826783700250676263362, 8.618966621189813626434174913816, 9.642727340101070443548067847258, 10.23827402727964604848387184191, 12.84688978513418512171462235404, 13.19069815363258335174666499645, 14.11125275639621900792162393084, 15.42382989853517912004077694458