L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.230 + 2.22i)5-s + (−0.866 − 0.499i)6-s + (0.432 − 0.749i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (2.04 − 0.912i)10-s + (0.151 − 0.0874i)11-s + 0.999i·12-s + (1.35 − 3.34i)13-s − 0.865·14-s + (0.912 + 2.04i)15-s + (−0.5 − 0.866i)16-s + (7.08 + 4.08i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.103 + 0.994i)5-s + (−0.353 − 0.204i)6-s + (0.163 − 0.283i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.645 − 0.288i)10-s + (0.0456 − 0.0263i)11-s + 0.288i·12-s + (0.375 − 0.926i)13-s − 0.231·14-s + (0.235 + 0.527i)15-s + (−0.125 − 0.216i)16-s + (1.71 + 0.991i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34670 - 0.344776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34670 - 0.344776i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.230 - 2.22i)T \) |
| 13 | \( 1 + (-1.35 + 3.34i)T \) |
good | 7 | \( 1 + (-0.432 + 0.749i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.151 + 0.0874i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-7.08 - 4.08i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.20 - 3.00i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.52 + 1.45i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.24 - 5.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.95iT - 31T^{2} \) |
| 37 | \( 1 + (0.879 + 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.08 - 4.08i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (7.94 + 4.58i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 2.48iT - 53T^{2} \) |
| 59 | \( 1 + (6.09 + 3.51i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.98 + 6.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.36 - 2.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.2 - 7.08i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 - 0.139T + 83T^{2} \) |
| 89 | \( 1 + (11.3 - 6.56i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.32 - 7.48i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.13118353556782443049588906516, −10.23760988476097393584608475391, −9.753058906806265222395738569318, −8.220753610828735787583180310809, −7.84353840453422090100801366456, −6.72828082226558426800795607855, −5.45048192293588107124121828585, −3.61222649285003970088733692278, −3.09196715857896594389506906372, −1.41903600236174698500313831235,
1.32726594854625284861379528582, 3.29790949815451012322332123848, 4.78316311468188923389726483347, 5.38249866225056436617686732349, 6.85340319574385666746698393450, 7.84842766838132594811917765035, 8.640201575566107001403152231311, 9.416920103408773198606021884345, 10.00721355458916193253340965143, 11.55104639175258602094611653026