| L(s) = 1 | − 2.07e4·2-s + 4.78e6·3-s − 1.08e8·4-s + 1.18e10·5-s − 9.90e10·6-s + 6.78e11·7-s + 1.33e13·8-s + 2.28e13·9-s − 2.45e14·10-s + 5.61e14·11-s − 5.17e14·12-s − 5.01e15·13-s − 1.40e16·14-s + 5.67e16·15-s − 2.18e17·16-s − 6.41e17·17-s − 4.73e17·18-s + 4.30e17·19-s − 1.28e18·20-s + 3.24e18·21-s − 1.16e19·22-s + 9.94e18·23-s + 6.38e19·24-s − 4.55e19·25-s + 1.03e20·26-s + 1.09e20·27-s − 7.33e19·28-s + ⋯ |
| L(s) = 1 | − 0.893·2-s + 0.577·3-s − 0.201·4-s + 0.869·5-s − 0.515·6-s + 0.377·7-s + 1.07·8-s + 0.333·9-s − 0.776·10-s + 0.446·11-s − 0.116·12-s − 0.353·13-s − 0.337·14-s + 0.501·15-s − 0.758·16-s − 0.924·17-s − 0.297·18-s + 0.123·19-s − 0.175·20-s + 0.218·21-s − 0.398·22-s + 0.178·23-s + 0.619·24-s − 0.244·25-s + 0.315·26-s + 0.192·27-s − 0.0761·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4.78e6T \) |
| 7 | \( 1 - 6.78e11T \) |
| good | 2 | \( 1 + 2.07e4T + 5.36e8T^{2} \) |
| 5 | \( 1 - 1.18e10T + 1.86e20T^{2} \) |
| 11 | \( 1 - 5.61e14T + 1.58e30T^{2} \) |
| 13 | \( 1 + 5.01e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + 6.41e17T + 4.81e35T^{2} \) |
| 19 | \( 1 - 4.30e17T + 1.21e37T^{2} \) |
| 23 | \( 1 - 9.94e18T + 3.09e39T^{2} \) |
| 29 | \( 1 + 2.76e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + 1.03e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 1.16e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 1.68e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 6.59e22T + 2.34e47T^{2} \) |
| 47 | \( 1 + 2.45e24T + 3.09e48T^{2} \) |
| 53 | \( 1 - 3.48e24T + 1.00e50T^{2} \) |
| 59 | \( 1 - 7.73e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 3.73e25T + 5.95e51T^{2} \) |
| 67 | \( 1 - 2.24e26T + 9.04e52T^{2} \) |
| 71 | \( 1 - 4.49e26T + 4.85e53T^{2} \) |
| 73 | \( 1 - 3.72e26T + 1.08e54T^{2} \) |
| 79 | \( 1 - 2.90e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 2.68e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + 3.21e28T + 3.40e56T^{2} \) |
| 97 | \( 1 + 5.82e28T + 4.13e57T^{2} \) |
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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
−11.18559138338481704902239646366, −9.869921768487045230652884255174, −9.190849605390401425853887479689, −8.146319890188359423212410082683, −6.90847511599089729557878145678, −5.25186504773814644117156403971, −3.94679478968185126646033970453, −2.20109171416240796008965709981, −1.41730852320042278439303853070, 0,
1.41730852320042278439303853070, 2.20109171416240796008965709981, 3.94679478968185126646033970453, 5.25186504773814644117156403971, 6.90847511599089729557878145678, 8.146319890188359423212410082683, 9.190849605390401425853887479689, 9.869921768487045230652884255174, 11.18559138338481704902239646366
