| L(s) = 1 | + (0.769 + 0.815i)2-s + (1.01 + 1.40i)3-s + (0.0432 − 0.742i)4-s + (−0.631 + 0.0738i)5-s + (−0.362 + 1.90i)6-s + (−4.36 − 2.19i)7-s + (2.35 − 1.97i)8-s + (−0.936 + 2.85i)9-s + (−0.546 − 0.458i)10-s + (0.510 + 1.18i)11-s + (1.08 − 0.693i)12-s + (1.03 + 0.245i)13-s + (−1.57 − 5.24i)14-s + (−0.745 − 0.811i)15-s + (1.94 + 0.227i)16-s + (−0.556 + 3.15i)17-s + ⋯ |
| L(s) = 1 | + (0.543 + 0.576i)2-s + (0.586 + 0.809i)3-s + (0.0216 − 0.371i)4-s + (−0.282 + 0.0330i)5-s + (−0.147 + 0.778i)6-s + (−1.65 − 0.828i)7-s + (0.832 − 0.698i)8-s + (−0.312 + 0.950i)9-s + (−0.172 − 0.144i)10-s + (0.153 + 0.356i)11-s + (0.313 − 0.200i)12-s + (0.287 + 0.0681i)13-s + (−0.419 − 1.40i)14-s + (−0.192 − 0.209i)15-s + (0.486 + 0.0568i)16-s + (−0.134 + 0.765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15638 + 0.515858i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15638 + 0.515858i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.01 - 1.40i)T \) |
| good | 2 | \( 1 + (-0.769 - 0.815i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (0.631 - 0.0738i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (4.36 + 2.19i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-0.510 - 1.18i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 0.245i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (0.556 - 3.15i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.503 + 2.85i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-6.50 + 3.26i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (1.59 - 5.34i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 0.727i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-7.60 + 2.76i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (7.70 - 8.16i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-2.00 + 2.69i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-0.240 + 0.158i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-2.40 + 4.15i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.95 - 6.84i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.202 + 3.47i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (2.70 + 9.02i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (7.54 + 6.33i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.548 + 0.460i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.72 - 1.82i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (4.81 + 5.10i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-8.45 + 7.09i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-8.87 - 1.03i)T + (94.3 + 22.3i)T^{2} \) |
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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
−14.72834764096863612461520016445, −13.50728254350982015800422651386, −12.96396048016441982490094126407, −10.87985658869133309752185230363, −10.06264331888020858493995857484, −9.097194545020761373683449623195, −7.32330043917111490038892523136, −6.26870606402303655619070981573, −4.59457409699569360824409147820, −3.45416464396420691677209834117,
2.66775315272243787726809911436, 3.64663700979544719031826958565, 5.95800940571217932627264850497, 7.28420207895398259879653252898, 8.565500286107929585582620255587, 9.639580846514419843366380778040, 11.51372337543663030499147048078, 12.23831603423786443932580254883, 13.15268749959259284685851560919, 13.66229337829676797567982645548
