| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s − 14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s − 21-s + 4·22-s + 24-s + 25-s + 26-s + 27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.049120203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.049120203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.30991169240681, −14.14807566070276, −13.72875689735240, −13.16969587468759, −12.61313640476589, −12.10617781413051, −11.66693664392311, −11.11644050106415, −10.29114953250944, −9.966470040422251, −9.434825152538390, −8.739492476918386, −8.433575845392386, −7.595693661608866, −6.863282954557089, −6.717167774968843, −5.978485467864241, −5.327557649201313, −4.843971092868776, −3.854413238599147, −3.701929540884194, −2.937934347106232, −2.290717204873097, −1.494278035588238, −0.8623481561895997,
0.8623481561895997, 1.494278035588238, 2.290717204873097, 2.937934347106232, 3.701929540884194, 3.854413238599147, 4.843971092868776, 5.327557649201313, 5.978485467864241, 6.717167774968843, 6.863282954557089, 7.595693661608866, 8.433575845392386, 8.739492476918386, 9.434825152538390, 9.966470040422251, 10.29114953250944, 11.11644050106415, 11.66693664392311, 12.10617781413051, 12.61313640476589, 13.16969587468759, 13.72875689735240, 14.14807566070276, 14.30991169240681
