| L(s) = 1 | + 2·3-s + 18·5-s − 7·7-s − 23·9-s + 11·11-s + 56·13-s + 36·15-s + 36·17-s + 28·19-s − 14·21-s − 180·23-s + 199·25-s − 100·27-s − 54·29-s + 334·31-s + 22·33-s − 126·35-s + 386·37-s + 112·39-s − 444·41-s + 316·43-s − 414·45-s + 402·47-s + 49·49-s + 72·51-s − 486·53-s + 198·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.60·5-s − 0.377·7-s − 0.851·9-s + 0.301·11-s + 1.19·13-s + 0.619·15-s + 0.513·17-s + 0.338·19-s − 0.145·21-s − 1.63·23-s + 1.59·25-s − 0.712·27-s − 0.345·29-s + 1.93·31-s + 0.116·33-s − 0.608·35-s + 1.71·37-s + 0.459·39-s − 1.69·41-s + 1.12·43-s − 1.37·45-s + 1.24·47-s + 1/7·49-s + 0.197·51-s − 1.25·53-s + 0.485·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.426957415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.426957415\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 334 T + p^{3} T^{2} \) |
| 37 | \( 1 - 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 444 T + p^{3} T^{2} \) |
| 43 | \( 1 - 316 T + p^{3} T^{2} \) |
| 47 | \( 1 - 402 T + p^{3} T^{2} \) |
| 53 | \( 1 + 486 T + p^{3} T^{2} \) |
| 59 | \( 1 - 282 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 176 T + p^{3} T^{2} \) |
| 71 | \( 1 - 324 T + p^{3} T^{2} \) |
| 73 | \( 1 - 800 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1144 T + p^{3} T^{2} \) |
| 83 | \( 1 + 468 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−9.521036393128536633060198224649, −8.580127876369033766494319854946, −7.949402702476332671590814436133, −6.49696288731283870632087152645, −6.05867196133069432753694204909, −5.40644271963252928783397492445, −4.00636501081945709607982175099, −2.95280177337331205499632189270, −2.10032209243304503302424070019, −0.960939649387840524989577139322,
0.960939649387840524989577139322, 2.10032209243304503302424070019, 2.95280177337331205499632189270, 4.00636501081945709607982175099, 5.40644271963252928783397492445, 6.05867196133069432753694204909, 6.49696288731283870632087152645, 7.949402702476332671590814436133, 8.580127876369033766494319854946, 9.521036393128536633060198224649
