L(s) = 1 | + (53.1 − 35.6i)2-s − 1.02e3·3-s + (1.55e3 − 3.79e3i)4-s + 2.12e4i·5-s + (−5.43e4 + 3.64e4i)6-s + 1.49e5i·7-s + (−5.27e4 − 2.56e5i)8-s + 5.13e5·9-s + (7.57e5 + 1.12e6i)10-s − 2.66e6·11-s + (−1.58e6 + 3.87e6i)12-s + 2.70e6i·13-s + (5.31e6 + 7.92e6i)14-s − 2.17e7i·15-s + (−1.19e7 − 1.17e7i)16-s + 8.65e6·17-s + ⋯ |
L(s) = 1 | + (0.830 − 0.557i)2-s − 1.40·3-s + (0.378 − 0.925i)4-s + 1.35i·5-s + (−1.16 + 0.781i)6-s + 1.26i·7-s + (−0.201 − 0.979i)8-s + 0.965·9-s + (0.757 + 1.12i)10-s − 1.50·11-s + (−0.531 + 1.29i)12-s + 0.560i·13-s + (0.706 + 1.05i)14-s − 1.90i·15-s + (−0.712 − 0.701i)16-s + 0.358·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.560108 + 0.686739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.560108 + 0.686739i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-53.1 + 35.6i)T \) |
good | 3 | \( 1 + 1.02e3T + 5.31e5T^{2} \) |
| 5 | \( 1 - 2.12e4iT - 2.44e8T^{2} \) |
| 7 | \( 1 - 1.49e5iT - 1.38e10T^{2} \) |
| 11 | \( 1 + 2.66e6T + 3.13e12T^{2} \) |
| 13 | \( 1 - 2.70e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 - 8.65e6T + 5.82e14T^{2} \) |
| 19 | \( 1 + 3.18e7T + 2.21e15T^{2} \) |
| 23 | \( 1 + 7.29e7iT - 2.19e16T^{2} \) |
| 29 | \( 1 - 7.54e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 + 1.62e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 - 2.33e9iT - 6.58e18T^{2} \) |
| 41 | \( 1 - 5.58e9T + 2.25e19T^{2} \) |
| 43 | \( 1 + 1.62e9T + 3.99e19T^{2} \) |
| 47 | \( 1 - 4.97e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 3.93e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 + 2.97e10T + 1.77e21T^{2} \) |
| 61 | \( 1 - 3.50e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 - 1.14e11T + 8.18e21T^{2} \) |
| 71 | \( 1 - 9.69e10iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 1.09e11T + 2.29e22T^{2} \) |
| 79 | \( 1 - 3.71e11iT - 5.90e22T^{2} \) |
| 83 | \( 1 - 7.22e10T + 1.06e23T^{2} \) |
| 89 | \( 1 - 6.12e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + 7.75e11T + 6.93e23T^{2} \) |
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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
−18.81461763075066599997908122843, −18.22124798194949566503234638193, −15.94482214800026257586318765458, −14.69558778455118194214055597719, −12.63589462113446218659905267382, −11.36155646245390177119777847827, −10.40889845416627236327278512965, −6.51067907453677788855868915565, −5.28236838001925575274718265820, −2.59763227495288490600146869405,
0.41144271372840403216473021353, 4.54476408260015526892293406301, 5.68848182162305454268147159134, 7.74624632230807061860690738421, 10.74667719762928713250988562525, 12.43450528493197705347948905557, 13.37408571708996496666275275845, 15.82365760402029108359865200032, 16.78226632858329819783003430960, 17.56631227192521988103311959977