L(s) = 1 | − 32i·2-s − 532. i·3-s − 1.02e3·4-s + (1.59e3 − 6.80e3i)5-s − 1.70e4·6-s + 2.44e4i·7-s + 3.27e4i·8-s − 1.06e5·9-s + (−2.17e5 − 5.10e4i)10-s − 6.69e5·11-s + 5.44e5i·12-s + 5.79e5i·13-s + 7.83e5·14-s + (−3.62e6 − 8.48e5i)15-s + 1.04e6·16-s − 7.85e6i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.26i·3-s − 0.5·4-s + (0.228 − 0.973i)5-s − 0.894·6-s + 0.550i·7-s + 0.353i·8-s − 0.598·9-s + (−0.688 − 0.161i)10-s − 1.25·11-s + 0.632i·12-s + 0.432i·13-s + 0.389·14-s + (−1.23 − 0.288i)15-s + 0.250·16-s − 1.34i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.147292 + 1.27385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.147292 + 1.27385i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 32iT \) |
| 5 | \( 1 + (-1.59e3 + 6.80e3i)T \) |
good | 3 | \( 1 + 532. iT - 1.77e5T^{2} \) |
| 7 | \( 1 - 2.44e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 6.69e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 5.79e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 7.85e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.81e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.97e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 2.02e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.02e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.02e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 3.44e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.43e9iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 1.38e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 2.89e8iT - 9.26e18T^{2} \) |
| 59 | \( 1 - 2.47e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.16e7T + 4.35e19T^{2} \) |
| 67 | \( 1 - 3.09e7iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 2.42e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.55e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 1.16e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.33e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 - 1.02e11T + 2.77e21T^{2} \) |
| 97 | \( 1 + 5.87e10iT - 7.15e21T^{2} \) |
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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
−18.00910224863678527329876879570, −16.14552477139428976875558225053, −13.73584877030380804469992736400, −12.80171197392832559384323944194, −11.71561964779402905227548977969, −9.396906711323708128250127533502, −7.70966936221151784144915269511, −5.31332013620488253374712914398, −2.29550708362711598235268447601, −0.68642359667407362647511778914,
3.55183640742035152665427837019, 5.47718555325092305754895097621, 7.59135748171717876928684857289, 9.803585997656866850439421241991, 10.75812317857135117298690778561, 13.50603399233296482936873549656, 14.99368393438504922038159646635, 15.79110148913278842434673570483, 17.25121688714027371339673642786, 18.61428196793718808981146953485