L(s) = 1 | + 128i·2-s − 2.93e3i·3-s − 1.63e4·4-s + (−1.60e5 − 6.84e4i)5-s + 3.75e5·6-s + 1.80e6i·7-s − 2.09e6i·8-s + 5.75e6·9-s + (8.76e6 − 2.05e7i)10-s + 8.15e7·11-s + 4.80e7i·12-s + 1.37e8i·13-s − 2.30e8·14-s + (−2.00e8 + 4.71e8i)15-s + 2.68e8·16-s + 2.67e9i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.774i·3-s − 0.5·4-s + (−0.919 − 0.392i)5-s + 0.547·6-s + 0.826i·7-s − 0.353i·8-s + 0.400·9-s + (0.277 − 0.650i)10-s + 1.26·11-s + 0.387i·12-s + 0.605i·13-s − 0.584·14-s + (−0.303 + 0.712i)15-s + 0.250·16-s + 1.58i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(1.23140 + 0.813725i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23140 + 0.813725i\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 128iT \) |
| 5 | \( 1 + (1.60e5 + 6.84e4i)T \) |
good | 3 | \( 1 + 2.93e3iT - 1.43e7T^{2} \) |
| 7 | \( 1 - 1.80e6iT - 4.74e12T^{2} \) |
| 11 | \( 1 - 8.15e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.37e8iT - 5.11e16T^{2} \) |
| 17 | \( 1 - 2.67e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 - 2.47e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.86e9iT - 2.66e20T^{2} \) |
| 29 | \( 1 + 1.20e11T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.95e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 2.27e11iT - 3.33e23T^{2} \) |
| 41 | \( 1 - 1.53e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.06e12iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 2.80e12iT - 1.20e25T^{2} \) |
| 53 | \( 1 + 5.32e12iT - 7.31e25T^{2} \) |
| 59 | \( 1 + 1.26e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.42e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 5.27e13iT - 2.46e27T^{2} \) |
| 71 | \( 1 - 2.86e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.30e14iT - 8.90e27T^{2} \) |
| 79 | \( 1 + 1.18e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 4.70e14iT - 6.11e28T^{2} \) |
| 89 | \( 1 - 5.97e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 9.88e14iT - 6.33e29T^{2} \) |
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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
−17.25554082294801727266517299776, −15.87334770776817871449344522479, −14.63711387076268010493899232265, −12.85795930391923849458733963015, −11.78375602974404385720227630205, −9.095686442129767313998788099008, −7.71899977220755141024084619124, −6.25751981155143702998880981447, −4.12878123722802406510614455531, −1.32059913438675733090261316158,
0.74201540605701600353620405410, 3.38854029011325120486098495479, 4.48958801595786704545969091624, 7.32798930922584770233204787129, 9.429881813399963121325632179836, 10.75290755944848218091269569943, 11.96713607389964167204245514023, 13.90187474797016841518242292694, 15.36067280563943669198091447655, 16.72372600272119938795179406531