L(s) = 1 | + (0.395 + 0.684i)2-s + (−4.5 + 7.79i)3-s + (15.6 − 27.1i)4-s + (52.0 + 90.2i)5-s − 7.11·6-s + (−7.12 + 129. i)7-s + 50.1·8-s + (−40.5 − 70.1i)9-s + (−41.1 + 71.3i)10-s + (248. − 430. i)11-s + (141. + 244. i)12-s − 206.·13-s + (−91.4 + 46.3i)14-s − 937.·15-s + (−482. − 835. i)16-s + (−31.5 + 54.6i)17-s + ⋯ |
L(s) = 1 | + (0.0698 + 0.121i)2-s + (−0.288 + 0.499i)3-s + (0.490 − 0.849i)4-s + (0.931 + 1.61i)5-s − 0.0807·6-s + (−0.0549 + 0.998i)7-s + 0.276·8-s + (−0.166 − 0.288i)9-s + (−0.130 + 0.225i)10-s + (0.620 − 1.07i)11-s + (0.283 + 0.490i)12-s − 0.338·13-s + (−0.124 + 0.0631i)14-s − 1.07·15-s + (−0.470 − 0.815i)16-s + (−0.0265 + 0.0459i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.45349 + 0.728115i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45349 + 0.728115i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.5 - 7.79i)T \) |
| 7 | \( 1 + (7.12 - 129. i)T \) |
good | 2 | \( 1 + (-0.395 - 0.684i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-52.0 - 90.2i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-248. + 430. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 206.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (31.5 - 54.6i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (661. + 1.14e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-97.2 - 168. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-3.76e3 + 6.51e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 4.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.19e3 - 3.80e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (8.89e3 - 1.54e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.75e3 - 3.03e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-5.31e3 - 9.20e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-6.63e3 + 1.14e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.56e4 - 2.71e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.97e4 + 3.42e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-5.64e4 - 9.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 3.03e4T + 8.58e9T^{2} \) |
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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
−17.37164416963853380741332962745, −15.71101599820335372435914677310, −14.79958755786609559070040512264, −13.91823846751875667812912224513, −11.50522390282443738026245206204, −10.59271850447395691654774767160, −9.390946790717921086278120514607, −6.58339595587481121657957889400, −5.73984646498827808965324466086, −2.59835994850349768932952320178,
1.55016034279135827785323409561, 4.60107374630584079775977856462, 6.74610556892610222850328816041, 8.351786685131296191529393069833, 10.05643030932958815490523016881, 12.09063940833894277972893176428, 12.76403088377064667794202476394, 13.88868287541100227381692211855, 16.17612161115586205137162099838, 17.21027849372280246176883621200