L(s) = 1 | + 1.56e3·2-s − 1.77e5·3-s − 5.94e6·4-s + 1.13e8·5-s − 2.77e8·6-s + 7.85e9·7-s − 2.24e10·8-s + 3.13e10·9-s + 1.77e11·10-s + 1.03e12·11-s + 1.05e12·12-s + 8.08e12·13-s + 1.22e13·14-s − 2.01e13·15-s + 1.48e13·16-s − 1.38e14·17-s + 4.90e13·18-s − 1.41e14·19-s − 6.75e14·20-s − 1.39e15·21-s + 1.61e15·22-s + 4.80e15·23-s + 3.97e15·24-s + 9.90e14·25-s + 1.26e16·26-s − 5.55e15·27-s − 4.66e16·28-s + ⋯ |
L(s) = 1 | + 0.539·2-s − 0.577·3-s − 0.708·4-s + 1.04·5-s − 0.311·6-s + 1.50·7-s − 0.922·8-s + 0.333·9-s + 0.561·10-s + 1.08·11-s + 0.409·12-s + 1.25·13-s + 0.810·14-s − 0.600·15-s + 0.210·16-s − 0.981·17-s + 0.179·18-s − 0.279·19-s − 0.737·20-s − 0.866·21-s + 0.588·22-s + 1.05·23-s + 0.532·24-s + 0.0831·25-s + 0.675·26-s − 0.192·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(12)\) |
\(\approx\) |
\(2.208741712\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.208741712\) |
\(L(\frac{25}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.77e5T \) |
good | 2 | \( 1 - 1.56e3T + 8.38e6T^{2} \) |
| 5 | \( 1 - 1.13e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 7.85e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.03e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 8.08e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.38e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 1.41e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 4.80e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 1.45e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 8.10e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.19e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 6.63e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.06e19T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.16e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 7.56e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 4.19e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 7.45e19T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.29e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.37e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.03e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 3.75e20T + 4.42e43T^{2} \) |
| 83 | \( 1 + 9.99e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 5.36e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.11e22T + 4.96e45T^{2} \) |
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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
−21.02346685635690036292227301379, −18.16776800142464036292479820873, −17.31607371478141726122240935026, −14.62677916772822452245415391676, −13.31147812131123570052965132983, −11.24589917709964709030554567062, −8.966255851812802516898269592285, −5.96245679305929442058441056383, −4.44630644229121449340527660913, −1.38131907346156944497222112369,
1.38131907346156944497222112369, 4.44630644229121449340527660913, 5.96245679305929442058441056383, 8.966255851812802516898269592285, 11.24589917709964709030554567062, 13.31147812131123570052965132983, 14.62677916772822452245415391676, 17.31607371478141726122240935026, 18.16776800142464036292479820873, 21.02346685635690036292227301379