| L(s) = 1 | + 6.04e4·2-s + 1.50e9·4-s + 5.19e10·5-s + 4.79e11·7-s − 3.86e13·8-s + 3.14e15·10-s − 2.28e16·11-s − 1.55e17·13-s + 2.89e16·14-s − 5.57e18·16-s + 8.05e17·17-s + 1.89e19·19-s + 7.83e19·20-s − 1.38e21·22-s + 2.34e21·23-s − 1.95e21·25-s − 9.42e21·26-s + 7.22e20·28-s − 4.26e22·29-s − 2.06e23·31-s − 2.54e23·32-s + 4.87e22·34-s + 2.49e22·35-s − 3.53e24·37-s + 1.14e24·38-s − 2.00e24·40-s + 2.76e24·41-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.702·4-s + 0.761·5-s + 0.0381·7-s − 0.388·8-s + 0.993·10-s − 1.65·11-s − 0.844·13-s + 0.0497·14-s − 1.20·16-s + 0.0682·17-s + 0.286·19-s + 0.534·20-s − 2.15·22-s + 1.83·23-s − 0.419·25-s − 1.10·26-s + 0.0267·28-s − 0.917·29-s − 1.58·31-s − 1.18·32-s + 0.0890·34-s + 0.0290·35-s − 1.74·37-s + 0.373·38-s − 0.295·40-s + 0.278·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 6.04e4T + 2.14e9T^{2} \) |
| 5 | \( 1 - 5.19e10T + 4.65e21T^{2} \) |
| 7 | \( 1 - 4.79e11T + 1.57e26T^{2} \) |
| 11 | \( 1 + 2.28e16T + 1.91e32T^{2} \) |
| 13 | \( 1 + 1.55e17T + 3.40e34T^{2} \) |
| 17 | \( 1 - 8.05e17T + 1.39e38T^{2} \) |
| 19 | \( 1 - 1.89e19T + 4.37e39T^{2} \) |
| 23 | \( 1 - 2.34e21T + 1.63e42T^{2} \) |
| 29 | \( 1 + 4.26e22T + 2.15e45T^{2} \) |
| 31 | \( 1 + 2.06e23T + 1.70e46T^{2} \) |
| 37 | \( 1 + 3.53e24T + 4.11e48T^{2} \) |
| 41 | \( 1 - 2.76e24T + 9.91e49T^{2} \) |
| 43 | \( 1 - 1.70e25T + 4.34e50T^{2} \) |
| 47 | \( 1 - 9.50e25T + 6.83e51T^{2} \) |
| 53 | \( 1 + 6.00e26T + 2.83e53T^{2} \) |
| 59 | \( 1 + 2.97e27T + 7.87e54T^{2} \) |
| 61 | \( 1 - 3.11e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 7.95e27T + 4.05e56T^{2} \) |
| 71 | \( 1 - 5.35e28T + 2.44e57T^{2} \) |
| 73 | \( 1 + 1.27e28T + 5.79e57T^{2} \) |
| 79 | \( 1 + 3.59e29T + 6.70e58T^{2} \) |
| 83 | \( 1 - 4.19e29T + 3.10e59T^{2} \) |
| 89 | \( 1 - 1.62e30T + 2.69e60T^{2} \) |
| 97 | \( 1 - 6.75e30T + 3.88e61T^{2} \) |
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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
−13.35019837856969012773689029932, −12.57867069339070787119514369793, −10.85608672854866864578640426297, −9.302581713946890115803615478929, −7.30796188901086139241117835674, −5.61538951461016745835405732066, −4.96186850631289931487325636953, −3.21928237657074592265994566312, −2.12960607619664671654134735745, 0,
2.12960607619664671654134735745, 3.21928237657074592265994566312, 4.96186850631289931487325636953, 5.61538951461016745835405732066, 7.30796188901086139241117835674, 9.302581713946890115803615478929, 10.85608672854866864578640426297, 12.57867069339070787119514369793, 13.35019837856969012773689029932
