| L(s) = 1 | + (−0.305 − 1.38i)2-s + (0.634 − 0.773i)3-s + (−1.81 + 0.843i)4-s + (2.47 − 0.751i)5-s + (−1.26 − 0.639i)6-s + (−0.467 − 0.0929i)7-s + (1.71 + 2.24i)8-s + (−0.195 − 0.980i)9-s + (−1.79 − 3.19i)10-s + (1.19 + 0.117i)11-s + (−0.498 + 1.93i)12-s + (1.52 − 5.03i)13-s + (0.0143 + 0.673i)14-s + (0.990 − 2.39i)15-s + (2.57 − 3.05i)16-s + (−0.970 − 2.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.215 − 0.976i)2-s + (0.366 − 0.446i)3-s + (−0.906 + 0.421i)4-s + (1.10 − 0.336i)5-s + (−0.514 − 0.261i)6-s + (−0.176 − 0.0351i)7-s + (0.607 + 0.794i)8-s + (−0.0650 − 0.326i)9-s + (−0.567 − 1.00i)10-s + (0.359 + 0.0354i)11-s + (−0.143 + 0.559i)12-s + (0.423 − 1.39i)13-s + (0.00383 + 0.180i)14-s + (0.255 − 0.617i)15-s + (0.644 − 0.764i)16-s + (−0.235 − 0.568i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.757523 - 1.26102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.757523 - 1.26102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.305 + 1.38i)T \) |
| 3 | \( 1 + (-0.634 + 0.773i)T \) |
| good | 5 | \( 1 + (-2.47 + 0.751i)T + (4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (0.467 + 0.0929i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.19 - 0.117i)T + (10.7 + 2.14i)T^{2} \) |
| 13 | \( 1 + (-1.52 + 5.03i)T + (-10.8 - 7.22i)T^{2} \) |
| 17 | \( 1 + (0.970 + 2.34i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.167 + 0.0893i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (2.91 - 4.36i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.0547 + 0.555i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (-3.59 - 3.59i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.95 + 3.66i)T + (-20.5 - 30.7i)T^{2} \) |
| 41 | \( 1 + (2.24 + 1.50i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-1.09 - 1.33i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (2.12 - 0.880i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (1.04 - 10.6i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (-1.51 - 4.99i)T + (-49.0 + 32.7i)T^{2} \) |
| 61 | \( 1 + (-8.12 - 6.66i)T + (11.9 + 59.8i)T^{2} \) |
| 67 | \( 1 + (5.90 + 4.84i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (0.500 - 2.51i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-12.7 + 2.53i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 4.85i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.18 - 2.21i)T + (-46.1 + 69.0i)T^{2} \) |
| 89 | \( 1 + (-8.18 - 12.2i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (9.33 + 9.33i)T + 97iT^{2} \) |
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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
−10.97115810126040098568446167610, −10.00463294055554137056706206198, −9.400333211458964799401407751724, −8.511079456818133222320307104806, −7.56155900963292716005640079820, −6.08990347348095570304116592843, −5.11714436395196946978115509884, −3.57855042249330806106167508073, −2.43806064701880799483407618182, −1.15911540002696738690717821004,
1.96708755180063679113110867392, 3.83097743881138166648643463073, 4.89660286795328154693875406962, 6.29235143664288386898402267690, 6.53435248880057083074042895078, 8.056901871177361673639741228558, 8.917318879856115029656443166267, 9.682887140397268168758953147043, 10.26381244585362325885086867763, 11.45099677319711428259214727202
