L(s) = 1 | + (1.11 + 1.93i)5-s + (1.36 − 0.366i)7-s + (3.87 + 2.23i)11-s + (4.09 + 1.09i)13-s + (−2.23 + 2.23i)17-s − 2i·19-s + (0.818 − 3.05i)23-s + (−2.5 + 4.33i)25-s + (−2.23 + 3.87i)29-s + (−2 − 3.46i)31-s + (2.23 + 2.23i)35-s + (−3 − 3i)37-s + (7.74 − 4.47i)41-s + (−1.09 − 4.09i)43-s + (2.45 + 9.16i)47-s + ⋯ |
L(s) = 1 | + (0.499 + 0.866i)5-s + (0.516 − 0.138i)7-s + (1.16 + 0.674i)11-s + (1.13 + 0.304i)13-s + (−0.542 + 0.542i)17-s − 0.458i·19-s + (0.170 − 0.636i)23-s + (−0.5 + 0.866i)25-s + (−0.415 + 0.719i)29-s + (−0.359 − 0.622i)31-s + (0.377 + 0.377i)35-s + (−0.493 − 0.493i)37-s + (1.20 − 0.698i)41-s + (−0.167 − 0.624i)43-s + (0.358 + 1.33i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.194326271\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.194326271\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.11 - 1.93i)T \) |
good | 7 | \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.87 - 2.23i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.09 - 1.09i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (2.23 - 2.23i)T - 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (-0.818 + 3.05i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.23 - 3.87i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 + (-7.74 + 4.47i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.09 + 4.09i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-2.45 - 9.16i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.23 + 2.23i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.47 - 7.74i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.366 - 1.36i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 4.47iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.19 - 3i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.16 + 2.45i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + (12.2 - 3.29i)T + (84.0 - 48.5i)T^{2} \) |
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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
−9.306489028361316967659268972967, −8.967759013186351450833448593481, −7.84541116236569198392343516934, −6.91934497188097447967240951761, −6.44546313509267821857791311904, −5.55785502739235535804502491653, −4.32580503661713705091055593264, −3.67267182813939585496174811113, −2.33298569583198620094557995712, −1.41878981870276321599380971754,
0.976486658054762368523879259690, 1.87003445935984427014040164447, 3.38271765178382891903760379679, 4.25729639419680885511472570736, 5.24742461797018143397772414459, 5.96562075081514679034929072956, 6.70205836803175755402379496388, 7.943393485501290368325421317343, 8.578401422843794311268921799162, 9.164960803391611544349103936345