L(s) = 1 | + (−0.458 + 0.888i)2-s + (0.866 + 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (0.989 − 0.142i)8-s + (0.5 + 0.866i)9-s + (−0.0950 − 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (−0.998 + 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s + i·27-s + ⋯ |
L(s) = 1 | + (−0.458 + 0.888i)2-s + (0.866 + 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (0.989 − 0.142i)8-s + (0.5 + 0.866i)9-s + (−0.0950 − 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (−0.998 + 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s + i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6829314885 + 1.583250328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6829314885 + 1.583250328i\) |
\(L(1)\) |
\(\approx\) |
\(0.8891604789 + 0.6505017673i\) |
\(L(1)\) |
\(\approx\) |
\(0.8891604789 + 0.6505017673i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.458 + 0.888i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.909 - 0.415i)T \) |
| 17 | \( 1 + (0.690 + 0.723i)T \) |
| 19 | \( 1 + (0.723 + 0.690i)T \) |
| 23 | \( 1 + (-0.945 - 0.327i)T \) |
| 29 | \( 1 + (0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.995 + 0.0950i)T \) |
| 37 | \( 1 + (-0.814 - 0.580i)T \) |
| 41 | \( 1 + (-0.841 + 0.540i)T \) |
| 43 | \( 1 + (0.989 - 0.142i)T \) |
| 47 | \( 1 + (0.998 + 0.0475i)T \) |
| 53 | \( 1 + (0.945 - 0.327i)T \) |
| 59 | \( 1 + (-0.888 + 0.458i)T \) |
| 61 | \( 1 + (-0.0475 + 0.998i)T \) |
| 67 | \( 1 + (0.998 - 0.0475i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.189 - 0.981i)T \) |
| 79 | \( 1 + (-0.928 + 0.371i)T \) |
| 83 | \( 1 + (0.755 + 0.654i)T \) |
| 89 | \( 1 + (0.235 + 0.971i)T \) |
| 97 | \( 1 + (0.989 - 0.142i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.40392638299662330425415567961, −17.52652688013733664808592700254, −17.167924615665221629606912497882, −16.021421095276631427679454360105, −15.53204158665754342135802179587, −14.31555477120093817462827875011, −13.98206005275702973581218635084, −13.42919331583832164563228676808, −12.44963703834174674051923786055, −12.01641622984494426139132962371, −11.53272839693563554881148196594, −10.24706258632831063831436451891, −9.92799946265557369777735507450, −9.12676240655476617805354843271, −8.62686461771412491672327059877, −7.66430985445583345911720879843, −7.370039003295703643200225845263, −6.4778020327326017343888221431, −5.18270714772733992486831817759, −4.4651052638647673886918834812, −3.5661297694681636135594604016, −2.860540409873883998957053350162, −2.31168339321247311207311420176, −1.42197288353834914353297227201, −0.578764022784411367546070859389,
0.98867933276835681729093178027, 1.96788190094036955813800044846, 2.87027441008109883208248274264, 3.850579579129611536286715900706, 4.50409216168114755186920219392, 5.340217399847700261304869749392, 5.96805112053769674400246762023, 6.99111048675737554345842226472, 7.711858854277595082202087735304, 8.15259330101665442134445224814, 8.827260372933342988901760507822, 9.65239796479798374560325376791, 10.26596293851238254199974032903, 10.47745814857616583165738645276, 11.92163616472043810267420226934, 12.57105981298175741271138801635, 13.632665904155758734806463345568, 14.0453145095747194816183197431, 14.6798332281056531370006812421, 15.2539834403494363786681018229, 15.900347394647514599843423477291, 16.47383819944530568209655939783, 17.18141584838607290221753260042, 17.85416706091995489360695120981, 18.65773200855303852770204727082