| L(s) = 1 | − 8.12e4·2-s + 4.30e7·3-s − 1.98e9·4-s + 5.17e11·5-s − 3.49e12·6-s − 3.34e13·7-s + 8.59e14·8-s + 1.85e15·9-s − 4.20e16·10-s − 8.29e16·11-s − 8.54e16·12-s − 3.31e18·13-s + 2.72e18·14-s + 2.22e19·15-s − 5.27e19·16-s + 2.58e20·17-s − 1.50e20·18-s + 1.73e21·19-s − 1.02e21·20-s − 1.44e21·21-s + 6.74e21·22-s + 3.47e22·23-s + 3.69e22·24-s + 1.51e23·25-s + 2.69e23·26-s + 7.97e22·27-s + 6.64e22·28-s + ⋯ |
| L(s) = 1 | − 0.876·2-s + 0.577·3-s − 0.231·4-s + 1.51·5-s − 0.506·6-s − 0.380·7-s + 1.07·8-s + 0.333·9-s − 1.33·10-s − 0.544·11-s − 0.133·12-s − 1.38·13-s + 0.333·14-s + 0.876·15-s − 0.715·16-s + 1.28·17-s − 0.292·18-s + 1.37·19-s − 0.350·20-s − 0.219·21-s + 0.477·22-s + 1.17·23-s + 0.623·24-s + 1.30·25-s + 1.21·26-s + 0.192·27-s + 0.0880·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(1.665681902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665681902\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4.30e7T \) |
| good | 2 | \( 1 + 8.12e4T + 8.58e9T^{2} \) |
| 5 | \( 1 - 5.17e11T + 1.16e23T^{2} \) |
| 7 | \( 1 + 3.34e13T + 7.73e27T^{2} \) |
| 11 | \( 1 + 8.29e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 3.31e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.58e20T + 4.02e40T^{2} \) |
| 19 | \( 1 - 1.73e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 3.47e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.23e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 4.83e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 6.61e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 4.65e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 4.73e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 2.09e26T + 1.51e55T^{2} \) |
| 53 | \( 1 - 2.04e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 5.20e28T + 2.74e58T^{2} \) |
| 61 | \( 1 - 4.57e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.36e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 1.81e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 9.48e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 5.17e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 4.73e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 2.98e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 6.56e32T + 3.65e65T^{2} \) |
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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
−17.95890401802934951099066158445, −16.73661530734739300430646520958, −14.31315554302090612309140912103, −13.07749128373282373052612590530, −10.00653831383892929429930472047, −9.421622027924170441613547079917, −7.52954939004550486519450736355, −5.25159371430610392723995867494, −2.64063185623969695195543043138, −1.05114857384417310432380055168,
1.05114857384417310432380055168, 2.64063185623969695195543043138, 5.25159371430610392723995867494, 7.52954939004550486519450736355, 9.421622027924170441613547079917, 10.00653831383892929429930472047, 13.07749128373282373052612590530, 14.31315554302090612309140912103, 16.73661530734739300430646520958, 17.95890401802934951099066158445
