| L(s) = 1 | + (0.130 − 0.303i)2-s + (−0.919 − 1.46i)3-s + (1.29 + 1.37i)4-s + (−0.175 − 3.01i)5-s + (−0.565 + 0.0868i)6-s + (0.00895 + 0.00212i)7-s + (1.20 − 0.439i)8-s + (−1.30 + 2.69i)9-s + (−0.935 − 0.340i)10-s + (2.97 + 1.95i)11-s + (0.825 − 3.16i)12-s + (−3.51 − 0.411i)13-s + (0.00181 − 0.00243i)14-s + (−4.25 + 3.02i)15-s + (−0.195 + 3.35i)16-s + (−3.09 + 2.59i)17-s + ⋯ |
| L(s) = 1 | + (0.0924 − 0.214i)2-s + (−0.530 − 0.847i)3-s + (0.648 + 0.687i)4-s + (−0.0784 − 1.34i)5-s + (−0.230 + 0.0354i)6-s + (0.00338 + 0.000802i)7-s + (0.426 − 0.155i)8-s + (−0.436 + 0.899i)9-s + (−0.295 − 0.107i)10-s + (0.896 + 0.589i)11-s + (0.238 − 0.914i)12-s + (−0.975 − 0.113i)13-s + (0.000485 − 0.000651i)14-s + (−1.09 + 0.781i)15-s + (−0.0488 + 0.837i)16-s + (−0.750 + 0.629i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.846923 - 0.452021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846923 - 0.452021i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.919 + 1.46i)T \) |
| good | 2 | \( 1 + (-0.130 + 0.303i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (0.175 + 3.01i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-0.00895 - 0.00212i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (-2.97 - 1.95i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (3.51 + 0.411i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (3.09 - 2.59i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-5.54 - 4.64i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (5.00 - 1.18i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (2.83 + 3.81i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 8.65i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.0241 - 0.137i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (1.42 + 3.31i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-1.09 + 0.551i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (-0.298 - 0.998i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-1.88 - 3.26i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.33 - 2.85i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (4.70 - 4.98i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-4.97 + 6.67i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (2.49 + 0.909i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.10 + 0.766i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (1.63 - 3.79i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (5.10 - 11.8i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (9.21 - 3.35i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.917 + 15.7i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−13.81143536469391868068683586988, −12.71706193241304738185863781509, −12.16133505838473958389218591618, −11.49913149793117974014289076221, −9.764524586114121117925198763590, −8.222451777548680832033552461591, −7.36044349230250704633235308870, −5.91255642175892932514337308411, −4.30109371743073919568010447266, −1.83644456382169672525381528837,
3.03942772065242751083855974703, 4.96569323884388465737336683797, 6.38243246391360133308305719778, 7.10946313657680006816509411154, 9.328605047690189409106834692436, 10.30505354273538480176066003786, 11.23473870665368326038546455492, 11.79522455761119462564423786125, 14.11716698898451532660846739802, 14.53226113236052501286460622226
