| L(s) = 1 | + 2.64e5·2-s − 1.16e9·3-s − 4.79e11·4-s − 6.30e13·5-s − 3.06e14·6-s − 4.59e16·7-s − 2.71e17·8-s + 1.35e18·9-s − 1.66e19·10-s − 1.42e20·11-s + 5.57e20·12-s + 2.23e20·13-s − 1.21e22·14-s + 7.32e22·15-s + 1.92e23·16-s + 3.05e23·17-s + 3.56e23·18-s − 1.47e25·19-s + 3.02e25·20-s + 5.33e25·21-s − 3.77e25·22-s − 5.75e26·23-s + 3.16e26·24-s + 2.15e27·25-s + 5.90e25·26-s − 1.57e27·27-s + 2.20e28·28-s + ⋯ |
| L(s) = 1 | + 0.356·2-s − 0.577·3-s − 0.873·4-s − 1.47·5-s − 0.205·6-s − 1.52·7-s − 0.667·8-s + 0.333·9-s − 0.526·10-s − 0.704·11-s + 0.504·12-s + 0.0424·13-s − 0.542·14-s + 0.853·15-s + 0.635·16-s + 0.309·17-s + 0.118·18-s − 1.70·19-s + 1.29·20-s + 0.878·21-s − 0.250·22-s − 1.60·23-s + 0.385·24-s + 1.18·25-s + 0.0151·26-s − 0.192·27-s + 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(20)\) |
\(\approx\) |
\(0.007480562113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007480562113\) |
| \(L(\frac{41}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.16e9T \) |
| good | 2 | \( 1 - 2.64e5T + 5.49e11T^{2} \) |
| 5 | \( 1 + 6.30e13T + 1.81e27T^{2} \) |
| 7 | \( 1 + 4.59e16T + 9.09e32T^{2} \) |
| 11 | \( 1 + 1.42e20T + 4.11e40T^{2} \) |
| 13 | \( 1 - 2.23e20T + 2.77e43T^{2} \) |
| 17 | \( 1 - 3.05e23T + 9.71e47T^{2} \) |
| 19 | \( 1 + 1.47e25T + 7.43e49T^{2} \) |
| 23 | \( 1 + 5.75e26T + 1.28e53T^{2} \) |
| 29 | \( 1 - 1.59e28T + 1.08e57T^{2} \) |
| 31 | \( 1 - 2.90e28T + 1.45e58T^{2} \) |
| 37 | \( 1 - 4.49e30T + 1.44e61T^{2} \) |
| 41 | \( 1 + 3.98e31T + 7.91e62T^{2} \) |
| 43 | \( 1 + 1.13e32T + 5.07e63T^{2} \) |
| 47 | \( 1 + 4.87e32T + 1.62e65T^{2} \) |
| 53 | \( 1 + 5.94e32T + 1.76e67T^{2} \) |
| 59 | \( 1 + 2.61e34T + 1.15e69T^{2} \) |
| 61 | \( 1 + 4.20e34T + 4.24e69T^{2} \) |
| 67 | \( 1 + 4.97e35T + 1.64e71T^{2} \) |
| 71 | \( 1 - 8.30e35T + 1.58e72T^{2} \) |
| 73 | \( 1 + 5.71e35T + 4.67e72T^{2} \) |
| 79 | \( 1 + 1.34e37T + 1.01e74T^{2} \) |
| 83 | \( 1 - 2.31e37T + 6.98e74T^{2} \) |
| 89 | \( 1 - 5.53e37T + 1.06e76T^{2} \) |
| 97 | \( 1 + 5.58e38T + 3.04e77T^{2} \) |
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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
−16.49005129973931558504073827616, −15.27553986095562255720854384855, −13.10662426979785944794974144451, −12.08736538840444288721523391180, −10.11272813802674521204998487875, −8.202488053188601532799446059622, −6.29219727325417041678329990124, −4.46148901211808124548251909046, −3.35303034384620127315404256382, −0.05062235783673003003673606429,
0.05062235783673003003673606429, 3.35303034384620127315404256382, 4.46148901211808124548251909046, 6.29219727325417041678329990124, 8.202488053188601532799446059622, 10.11272813802674521204998487875, 12.08736538840444288721523391180, 13.10662426979785944794974144451, 15.27553986095562255720854384855, 16.49005129973931558504073827616
