| L(s) = 1 | + 5.46·2-s + 21.8·4-s + 0.199·5-s + 75.5·8-s + 1.08·10-s + 28.4·11-s + 32.5·13-s + 237.·16-s − 115.·17-s + 21.1·19-s + 4.34·20-s + 155.·22-s + 93.7·23-s − 124.·25-s + 177.·26-s + 231.·29-s + 281.·31-s + 695.·32-s − 630.·34-s − 146.·37-s + 115.·38-s + 15.0·40-s − 111.·41-s + 392.·43-s + 620.·44-s + 512.·46-s + 273.·47-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.72·4-s + 0.0178·5-s + 3.33·8-s + 0.0343·10-s + 0.779·11-s + 0.695·13-s + 3.71·16-s − 1.64·17-s + 0.255·19-s + 0.0486·20-s + 1.50·22-s + 0.850·23-s − 0.999·25-s + 1.34·26-s + 1.48·29-s + 1.63·31-s + 3.84·32-s − 3.18·34-s − 0.651·37-s + 0.493·38-s + 0.0594·40-s − 0.422·41-s + 1.39·43-s + 2.12·44-s + 1.64·46-s + 0.847·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.119175778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.119175778\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.46T + 8T^{2} \) |
| 5 | \( 1 - 0.199T + 125T^{2} \) |
| 11 | \( 1 - 28.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 231.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 281.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 392.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 273.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 340.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 696.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 87.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 88.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 803.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 364.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 921.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 211.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 845.T + 9.12e5T^{2} \) |
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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
−9.273671400522211943743129977485, −8.248025003291125509669939959750, −7.15700987497031083896530075198, −6.44055063108547925135865393245, −5.97685460593182160109825110399, −4.74656406976674737890711139423, −4.30504271645170610062690929716, −3.30956944547911110256672801830, −2.41493247290483231253852739880, −1.27134869292084255218831776704,
1.27134869292084255218831776704, 2.41493247290483231253852739880, 3.30956944547911110256672801830, 4.30504271645170610062690929716, 4.74656406976674737890711139423, 5.97685460593182160109825110399, 6.44055063108547925135865393245, 7.15700987497031083896530075198, 8.248025003291125509669939959750, 9.273671400522211943743129977485
