| L(s) = 1 | + 58.7·5-s − 167·7-s + 764.·11-s − 235·13-s + 176.·17-s − 1.36e3·19-s + 2.41e3·23-s + 331·25-s + 470.·29-s − 3.50e3·31-s − 9.81e3·35-s + 1.31e4·37-s + 9.40e3·41-s − 104·43-s + 2.05e4·47-s + 1.10e4·49-s − 1.05e3·53-s + 4.49e4·55-s − 3.07e4·59-s − 7.39e3·61-s − 1.38e4·65-s − 3.88e4·67-s + 2.46e3·71-s + 5.46e3·73-s − 1.27e5·77-s + 8.29e4·79-s + 1.32e4·83-s + ⋯ |
| L(s) = 1 | + 1.05·5-s − 1.28·7-s + 1.90·11-s − 0.385·13-s + 0.148·17-s − 0.864·19-s + 0.950·23-s + 0.105·25-s + 0.103·29-s − 0.654·31-s − 1.35·35-s + 1.57·37-s + 0.873·41-s − 0.00857·43-s + 1.35·47-s + 0.659·49-s − 0.0517·53-s + 2.00·55-s − 1.14·59-s − 0.254·61-s − 0.405·65-s − 1.05·67-s + 0.0581·71-s + 0.120·73-s − 2.45·77-s + 1.49·79-s + 0.211·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.466844109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466844109\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 58.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 167T + 1.68e4T^{2} \) |
| 11 | \( 1 - 764.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 235T + 3.71e5T^{2} \) |
| 17 | \( 1 - 176.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 470.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.31e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 104T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.05e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.39e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.46e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.46e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.96e4T + 8.58e9T^{2} \) |
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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
−10.14210675039774532664000174513, −9.298263954365579670808783056750, −9.051749580463913004621771850841, −7.34763080616690381679047801120, −6.35457721325133445967781886012, −5.99218458246263514321199426307, −4.44214835854310044298646324527, −3.33535947519520098177387795162, −2.11232860695133741422162572760, −0.831127351957015006441766649926,
0.831127351957015006441766649926, 2.11232860695133741422162572760, 3.33535947519520098177387795162, 4.44214835854310044298646324527, 5.99218458246263514321199426307, 6.35457721325133445967781886012, 7.34763080616690381679047801120, 9.051749580463913004621771850841, 9.298263954365579670808783056750, 10.14210675039774532664000174513
