| L(s) = 1 | + 3·3-s + 14·5-s − 7·7-s + 9·9-s + 4·11-s + 54·13-s + 42·15-s − 14·17-s + 92·19-s − 21·21-s − 152·23-s + 71·25-s + 27·27-s − 106·29-s − 144·31-s + 12·33-s − 98·35-s + 158·37-s + 162·39-s − 390·41-s − 508·43-s + 126·45-s − 528·47-s + 49·49-s − 42·51-s + 606·53-s + 56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.25·5-s − 0.377·7-s + 1/3·9-s + 0.109·11-s + 1.15·13-s + 0.722·15-s − 0.199·17-s + 1.11·19-s − 0.218·21-s − 1.37·23-s + 0.567·25-s + 0.192·27-s − 0.678·29-s − 0.834·31-s + 0.0633·33-s − 0.473·35-s + 0.702·37-s + 0.665·39-s − 1.48·41-s − 1.80·43-s + 0.417·45-s − 1.63·47-s + 1/7·49-s − 0.115·51-s + 1.57·53-s + 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.110947014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.110947014\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 508 T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 - 606 T + p^{3} T^{2} \) |
| 59 | \( 1 + 364 T + p^{3} T^{2} \) |
| 61 | \( 1 - 678 T + p^{3} T^{2} \) |
| 67 | \( 1 - 844 T + p^{3} T^{2} \) |
| 71 | \( 1 + 8 T + p^{3} T^{2} \) |
| 73 | \( 1 + 422 T + p^{3} T^{2} \) |
| 79 | \( 1 - 384 T + p^{3} T^{2} \) |
| 83 | \( 1 + 548 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1502 T + p^{3} T^{2} \) |
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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
−13.61045143602081356968925505363, −13.17280224070887644369620931583, −11.61953290530631660942745782021, −10.16029493914475941599106264168, −9.442543966698980303383585798568, −8.266563216992068196997599658452, −6.66830121119897235013143547927, −5.52983004146289528330926984739, −3.53869853400353661775319926359, −1.80472185195327873760292891635,
1.80472185195327873760292891635, 3.53869853400353661775319926359, 5.52983004146289528330926984739, 6.66830121119897235013143547927, 8.266563216992068196997599658452, 9.442543966698980303383585798568, 10.16029493914475941599106264168, 11.61953290530631660942745782021, 13.17280224070887644369620931583, 13.61045143602081356968925505363
