| L(s) = 1 | − 121.·2-s − 1.59e6·3-s − 1.34e8·4-s − 1.07e9·5-s + 1.93e8·6-s − 1.13e10·7-s + 3.25e10·8-s + 2.54e12·9-s + 1.30e11·10-s − 5.05e13·11-s + 2.13e14·12-s + 1.21e15·13-s + 1.37e12·14-s + 1.71e15·15-s + 1.80e16·16-s + 7.00e16·17-s − 3.07e14·18-s + 2.51e16·19-s + 1.44e17·20-s + 1.81e16·21-s + 6.12e15·22-s − 3.40e18·23-s − 5.18e16·24-s − 6.28e18·25-s − 1.46e17·26-s − 4.05e18·27-s + 1.52e18·28-s + ⋯ |
| L(s) = 1 | − 0.0104·2-s − 0.577·3-s − 0.999·4-s − 0.394·5-s + 0.00603·6-s − 0.0444·7-s + 0.0209·8-s + 0.333·9-s + 0.00412·10-s − 0.441·11-s + 0.577·12-s + 1.11·13-s + 0.000464·14-s + 0.228·15-s + 0.999·16-s + 1.71·17-s − 0.00348·18-s + 0.136·19-s + 0.394·20-s + 0.0256·21-s + 0.00461·22-s − 1.40·23-s − 0.0120·24-s − 0.844·25-s − 0.0116·26-s − 0.192·27-s + 0.0444·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(0.9980317501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9980317501\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.59e6T \) |
| good | 2 | \( 1 + 121.T + 1.34e8T^{2} \) |
| 5 | \( 1 + 1.07e9T + 7.45e18T^{2} \) |
| 7 | \( 1 + 1.13e10T + 6.57e22T^{2} \) |
| 11 | \( 1 + 5.05e13T + 1.31e28T^{2} \) |
| 13 | \( 1 - 1.21e15T + 1.19e30T^{2} \) |
| 17 | \( 1 - 7.00e16T + 1.66e33T^{2} \) |
| 19 | \( 1 - 2.51e16T + 3.36e34T^{2} \) |
| 23 | \( 1 + 3.40e18T + 5.84e36T^{2} \) |
| 29 | \( 1 - 5.74e19T + 3.05e39T^{2} \) |
| 31 | \( 1 - 1.08e20T + 1.84e40T^{2} \) |
| 37 | \( 1 + 1.90e21T + 2.19e42T^{2} \) |
| 41 | \( 1 - 6.66e21T + 3.50e43T^{2} \) |
| 43 | \( 1 - 1.03e21T + 1.26e44T^{2} \) |
| 47 | \( 1 - 6.43e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.13e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 1.07e24T + 6.50e47T^{2} \) |
| 61 | \( 1 - 8.16e23T + 1.59e48T^{2} \) |
| 67 | \( 1 - 1.68e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 3.18e24T + 9.63e49T^{2} \) |
| 73 | \( 1 - 2.09e25T + 2.04e50T^{2} \) |
| 79 | \( 1 + 8.88e22T + 1.72e51T^{2} \) |
| 83 | \( 1 + 6.50e25T + 6.53e51T^{2} \) |
| 89 | \( 1 - 3.38e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 4.16e26T + 4.39e53T^{2} \) |
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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
−18.99111174389490532947712456263, −17.75290237180002296568425167393, −16.01085138427895718668024794883, −13.88711071268146075792173720516, −12.17052150369082891496338677350, −10.13118835683720197053931014935, −8.105919284656919411437338750971, −5.63613204169456285987627106693, −3.85754340835505704010694964980, −0.799486610718465007490355893782,
0.799486610718465007490355893782, 3.85754340835505704010694964980, 5.63613204169456285987627106693, 8.105919284656919411437338750971, 10.13118835683720197053931014935, 12.17052150369082891496338677350, 13.88711071268146075792173720516, 16.01085138427895718668024794883, 17.75290237180002296568425167393, 18.99111174389490532947712456263
