L(s) = 1 | + (1.71 − 0.272i)3-s + (0.119 + 0.207i)5-s + (2.85 − 0.931i)9-s + (2.56 − 4.43i)11-s + (−2.44 − 4.23i)13-s + (0.260 + 0.321i)15-s − 3.70·17-s − 3.66·19-s + (−3.71 − 6.42i)23-s + (2.47 − 4.28i)25-s + (4.62 − 2.36i)27-s + (−1.73 + 3.00i)29-s + (−0.358 − 0.621i)31-s + (3.17 − 8.28i)33-s + 4.60·37-s + ⋯ |
L(s) = 1 | + (0.987 − 0.157i)3-s + (0.0534 + 0.0926i)5-s + (0.950 − 0.310i)9-s + (0.772 − 1.33i)11-s + (−0.677 − 1.17i)13-s + (0.0673 + 0.0830i)15-s − 0.898·17-s − 0.839·19-s + (−0.773 − 1.34i)23-s + (0.494 − 0.856i)25-s + (0.890 − 0.455i)27-s + (−0.321 + 0.557i)29-s + (−0.0644 − 0.111i)31-s + (0.552 − 1.44i)33-s + 0.756·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.244473397\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.244473397\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 + 0.272i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.119 - 0.207i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.56 + 4.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 + 4.23i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 + (3.71 + 6.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.73 - 3.00i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.358 + 0.621i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.24 - 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.942T + 53T^{2} \) |
| 59 | \( 1 + (-3.78 - 6.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.75 + 4.77i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.330 - 0.571i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 3.66T + 73T^{2} \) |
| 79 | \( 1 + (-3.11 + 5.39i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.85 + 8.40i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + (8.57 - 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.912698392142894322547493177589, −8.367713005037320592956038453565, −7.81209842223105801978990527155, −6.59650265739066379166333315535, −6.22484627577271810285491863008, −4.83499915051380659819745496431, −3.97419985136907493367635670964, −3.00620900138416945465463444778, −2.24984536710322698267771634158, −0.71687220000516924054773501633,
1.79663080144774905293221094850, 2.25506716959470395903380307495, 3.81511755197023790480399028953, 4.26150615003033586097701064175, 5.19394209981634512370633615092, 6.64928418357528421919849784257, 7.08128039353318888544255569866, 7.901957330329553626563187086985, 8.908136700471810768468211366553, 9.423035084649114526932334616913