| L(s) = 1 | + (−0.597 + 0.802i)2-s + (−0.286 − 0.957i)4-s + (−1.23 − 0.810i)5-s + (0.694 + 0.736i)7-s + (0.939 + 0.342i)8-s + (1.38 − 0.504i)10-s + (−0.261 − 0.131i)11-s + (1.36 − 3.16i)13-s + (−1.00 + 0.117i)14-s + (−0.835 + 0.549i)16-s + (1.92 + 1.61i)17-s + (3.04 − 2.55i)19-s + (−0.423 + 1.41i)20-s + (0.261 − 0.131i)22-s + (3.54 − 3.75i)23-s + ⋯ |
| L(s) = 1 | + (−0.422 + 0.567i)2-s + (−0.143 − 0.478i)4-s + (−0.551 − 0.362i)5-s + (0.262 + 0.278i)7-s + (0.332 + 0.120i)8-s + (0.438 − 0.159i)10-s + (−0.0787 − 0.0395i)11-s + (0.378 − 0.876i)13-s + (−0.268 + 0.0314i)14-s + (−0.208 + 0.137i)16-s + (0.467 + 0.392i)17-s + (0.699 − 0.586i)19-s + (−0.0946 + 0.316i)20-s + (0.0556 − 0.0279i)22-s + (0.738 − 0.782i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05632 - 0.0556693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05632 - 0.0556693i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.597 - 0.802i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.23 + 0.810i)T + (1.98 + 4.59i)T^{2} \) |
| 7 | \( 1 + (-0.694 - 0.736i)T + (-0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (0.261 + 0.131i)T + (6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 3.16i)T + (-8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (-1.92 - 1.61i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.04 + 2.55i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-3.54 + 3.75i)T + (-1.33 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-9.88 - 1.15i)T + (28.2 + 6.68i)T^{2} \) |
| 31 | \( 1 + (-1.32 - 0.313i)T + (27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-1.03 + 5.87i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (2.84 + 3.82i)T + (-11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.689 - 11.8i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (-2.71 + 0.643i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (1.31 - 2.28i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.159 - 0.0801i)T + (35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (2.93 - 9.80i)T + (-50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (-8.02 + 0.938i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (13.2 - 4.81i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.12 + 0.411i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.40 + 7.26i)T + (-22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-7.45 + 10.0i)T + (-23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (10.6 + 3.88i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.36 - 5.50i)T + (38.4 - 89.0i)T^{2} \) |
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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
−10.79630238900400378258848098587, −10.06039326076103611157705880021, −8.868994912155817209705577579381, −8.294097176201426993586432622798, −7.49793293559972456176160492986, −6.38476695899871873714812879272, −5.35897055535811578749083294307, −4.41820648419390453636466779924, −2.90162926322336807549095627780, −0.902307211385319065920806252418,
1.31961326879655994851328397218, 2.97489980856010772748983701915, 3.95387321058844390816504491664, 5.12429566699013676213865411717, 6.61998936178542233306629247972, 7.51185214359797444848488678932, 8.292525773994686961760538965984, 9.340601183786523634580414284153, 10.13489816475434818343347377766, 11.07788660495575294201295721141
