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
−9.423035084649114526932334616913, −8.908136700471810768468211366553, −7.901957330329553626563187086985, −7.08128039353318888544255569866, −6.64928418357528421919849784257, −5.19394209981634512370633615092, −4.26150615003033586097701064175, −3.81511755197023790480399028953, −2.25506716959470395903380307495, −1.79663080144774905293221094850,
0.71687220000516924054773501633, 2.24984536710322698267771634158, 3.00620900138416945465463444778, 3.97419985136907493367635670964, 4.83499915051380659819745496431, 6.22484627577271810285491863008, 6.59650265739066379166333315535, 7.81209842223105801978990527155, 8.367713005037320592956038453565, 8.912698392142894322547493177589