| L(s) = 1 | + 3-s + 3·5-s − 7-s + 9-s − 3·11-s + 13-s + 3·15-s + 17-s + 5·19-s − 21-s + 3·23-s + 4·25-s + 27-s + 6·29-s + 10·31-s − 3·33-s − 3·35-s + 4·37-s + 39-s − 9·41-s − 43-s + 3·45-s + 6·47-s + 49-s + 51-s − 6·53-s − 9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.774·15-s + 0.242·17-s + 1.14·19-s − 0.218·21-s + 0.625·23-s + 4/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.522·33-s − 0.507·35-s + 0.657·37-s + 0.160·39-s − 1.40·41-s − 0.152·43-s + 0.447·45-s + 0.875·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.099604209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.099604209\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.63357509622585, −14.91846879318389, −14.23980857698047, −13.62760431023907, −13.58409424401539, −13.04584267742040, −12.23246415057702, −11.88286542352941, −10.87776038720505, −10.37709854692089, −9.954046906539514, −9.459532722545872, −8.962387442369939, −8.210904182498354, −7.757179924516849, −6.982268710856146, −6.310833138119245, −5.917523189522793, −5.005744905172483, −4.763100324346999, −3.550501927231247, −2.907855981093421, −2.512660983853667, −1.570627266971556, −0.8221888680149231,
0.8221888680149231, 1.570627266971556, 2.512660983853667, 2.907855981093421, 3.550501927231247, 4.763100324346999, 5.005744905172483, 5.917523189522793, 6.310833138119245, 6.982268710856146, 7.757179924516849, 8.210904182498354, 8.962387442369939, 9.459532722545872, 9.954046906539514, 10.37709854692089, 10.87776038720505, 11.88286542352941, 12.23246415057702, 13.04584267742040, 13.58409424401539, 13.62760431023907, 14.23980857698047, 14.91846879318389, 15.63357509622585
