| L(s) = 1 | − 4.38·3-s + 25·5-s − 223.·9-s + 579.·11-s + 426.·13-s − 109.·15-s + 2.01e3·17-s + 402.·19-s + 1.51e3·23-s + 625·25-s + 2.04e3·27-s − 7.78e3·29-s + 4.70e3·31-s − 2.53e3·33-s + 1.77e3·37-s − 1.86e3·39-s + 7.35e3·41-s − 8.56e3·43-s − 5.59e3·45-s − 3.74e3·47-s − 8.80e3·51-s − 3.35e4·53-s + 1.44e4·55-s − 1.76e3·57-s + 1.13e4·59-s + 2.49e4·61-s + 1.06e4·65-s + ⋯ |
| L(s) = 1 | − 0.281·3-s + 0.447·5-s − 0.920·9-s + 1.44·11-s + 0.699·13-s − 0.125·15-s + 1.68·17-s + 0.255·19-s + 0.597·23-s + 0.200·25-s + 0.540·27-s − 1.71·29-s + 0.878·31-s − 0.405·33-s + 0.212·37-s − 0.196·39-s + 0.683·41-s − 0.706·43-s − 0.411·45-s − 0.247·47-s − 0.474·51-s − 1.63·53-s + 0.645·55-s − 0.0718·57-s + 0.422·59-s + 0.859·61-s + 0.312·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.656221786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.656221786\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4.38T + 243T^{2} \) |
| 11 | \( 1 - 579.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 426.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 402.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.74e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.50e4T + 8.58e9T^{2} \) |
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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
−9.311777039198551114208555170402, −8.537123873069959178545094519868, −7.60192655835063117261823555718, −6.50390358245507566598635454898, −5.89477845269386338918252439055, −5.11975422602787745442066778086, −3.80674750265579950915729194445, −3.04645909342565458496681575559, −1.61181318770494918014945942837, −0.77685159906677371088932789037,
0.77685159906677371088932789037, 1.61181318770494918014945942837, 3.04645909342565458496681575559, 3.80674750265579950915729194445, 5.11975422602787745442066778086, 5.89477845269386338918252439055, 6.50390358245507566598635454898, 7.60192655835063117261823555718, 8.537123873069959178545094519868, 9.311777039198551114208555170402
