L(s) = 1 | + (−0.826 + 0.563i)2-s + (1.73 − 0.0218i)3-s + (0.365 − 0.930i)4-s + (−0.399 − 1.75i)5-s + (−1.41 + 0.993i)6-s + (−2.08 + 1.62i)7-s + (0.222 + 0.974i)8-s + (2.99 − 0.0755i)9-s + (1.31 + 1.22i)10-s + (−4.36 − 2.10i)11-s + (0.612 − 1.62i)12-s + (−5.61 + 3.82i)13-s + (0.808 − 2.51i)14-s + (−0.730 − 3.02i)15-s + (−0.733 − 0.680i)16-s + (−2.26 − 5.77i)17-s + ⋯ |
L(s) = 1 | + (−0.584 + 0.398i)2-s + (0.999 − 0.0125i)3-s + (0.182 − 0.465i)4-s + (−0.178 − 0.783i)5-s + (−0.579 + 0.405i)6-s + (−0.788 + 0.614i)7-s + (0.0786 + 0.344i)8-s + (0.999 − 0.0251i)9-s + (0.416 + 0.386i)10-s + (−1.31 − 0.633i)11-s + (0.176 − 0.467i)12-s + (−1.55 + 1.06i)13-s + (0.216 − 0.673i)14-s + (−0.188 − 0.781i)15-s + (−0.183 − 0.170i)16-s + (−0.550 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.163496 - 0.406457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.163496 - 0.406457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 3 | \( 1 + (-1.73 + 0.0218i)T \) |
| 7 | \( 1 + (2.08 - 1.62i)T \) |
good | 5 | \( 1 + (0.399 + 1.75i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (4.36 + 2.10i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (5.61 - 3.82i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (2.26 + 5.77i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (3.35 + 5.80i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.92 - 3.66i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-3.39 - 0.511i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (1.14 + 1.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (11.1 + 1.67i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (1.47 + 0.455i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (9.76 - 3.01i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (1.96 - 1.33i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-10.0 + 1.52i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-0.0711 + 0.0219i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-0.870 - 2.21i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-2.79 - 4.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.865 + 1.08i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.690 - 9.21i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-3.43 + 5.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.1 + 8.29i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (-4.38 - 2.99i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-1.66 - 2.88i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.518865266391621571061941391428, −8.869011490449843194102312775176, −8.464083745250931201108910027328, −7.17383885570956505803917558335, −6.89520260534738341786277640141, −5.21600685474511570654597859077, −4.69358754473127632120993824949, −2.96231907949480280094393957908, −2.25772663742386960489604183965, −0.20135358769888503024266606309,
2.06135364312557487936829194463, 2.92923348762518166211831535203, 3.70355831647707230598993677041, 4.95105266025634263625083052079, 6.63798525130719073805111472054, 7.23850843419582515149634144613, 8.013091643281680884539535467921, 8.661138552878423161355509279846, 10.03166662110521289779906257782, 10.33163677127167022014783981373