| L(s) = 1 | + (−0.992 − 0.596i)2-s + (0.370 − 0.928i)3-s + (−0.308 − 0.582i)4-s + (−0.793 − 0.0863i)5-s + (−0.921 + 0.700i)6-s + (−4.18 − 1.41i)7-s + (−0.166 + 3.07i)8-s + (−0.725 − 0.687i)9-s + (0.736 + 0.559i)10-s + (0.744 + 1.09i)11-s + (−0.655 + 0.0712i)12-s + (2.28 − 2.16i)13-s + (3.31 + 3.90i)14-s + (−0.374 + 0.705i)15-s + (1.26 − 1.85i)16-s + (−7.48 + 2.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.701 − 0.422i)2-s + (0.213 − 0.536i)3-s + (−0.154 − 0.291i)4-s + (−0.355 − 0.0386i)5-s + (−0.376 + 0.286i)6-s + (−1.58 − 0.533i)7-s + (−0.0589 + 1.08i)8-s + (−0.241 − 0.229i)9-s + (0.232 + 0.176i)10-s + (0.224 + 0.330i)11-s + (−0.189 + 0.0205i)12-s + (0.633 − 0.600i)13-s + (0.885 + 1.04i)14-s + (−0.0965 + 0.182i)15-s + (0.315 − 0.464i)16-s + (−1.81 + 0.611i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000226254 + 0.412620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000226254 + 0.412620i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.370 + 0.928i)T \) |
| 59 | \( 1 + (7.67 - 0.392i)T \) |
| good | 2 | \( 1 + (0.992 + 0.596i)T + (0.936 + 1.76i)T^{2} \) |
| 5 | \( 1 + (0.793 + 0.0863i)T + (4.88 + 1.07i)T^{2} \) |
| 7 | \( 1 + (4.18 + 1.41i)T + (5.57 + 4.23i)T^{2} \) |
| 11 | \( 1 + (-0.744 - 1.09i)T + (-4.07 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-2.28 + 2.16i)T + (0.703 - 12.9i)T^{2} \) |
| 17 | \( 1 + (7.48 - 2.52i)T + (13.5 - 10.2i)T^{2} \) |
| 19 | \( 1 + (-0.837 + 5.10i)T + (-18.0 - 6.06i)T^{2} \) |
| 23 | \( 1 + (-0.662 + 2.38i)T + (-19.7 - 11.8i)T^{2} \) |
| 29 | \( 1 + (-6.94 + 4.17i)T + (13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (0.686 + 4.18i)T + (-29.3 + 9.89i)T^{2} \) |
| 37 | \( 1 + (-0.435 - 8.04i)T + (-36.7 + 4.00i)T^{2} \) |
| 41 | \( 1 + (1.62 + 5.83i)T + (-35.1 + 21.1i)T^{2} \) |
| 43 | \( 1 + (-6.39 + 9.42i)T + (-15.9 - 39.9i)T^{2} \) |
| 47 | \( 1 + (-3.69 + 0.402i)T + (45.9 - 10.1i)T^{2} \) |
| 53 | \( 1 + (1.10 - 0.839i)T + (14.1 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.88 - 5.34i)T + (28.5 + 53.8i)T^{2} \) |
| 67 | \( 1 + (0.0694 - 1.28i)T + (-66.6 - 7.24i)T^{2} \) |
| 71 | \( 1 + (-0.433 + 0.0471i)T + (69.3 - 15.2i)T^{2} \) |
| 73 | \( 1 + (1.39 + 1.64i)T + (-11.8 + 72.0i)T^{2} \) |
| 79 | \( 1 + (1.73 + 4.36i)T + (-57.3 + 54.3i)T^{2} \) |
| 83 | \( 1 + (3.34 + 1.54i)T + (53.7 + 63.2i)T^{2} \) |
| 89 | \( 1 + (5.98 - 3.60i)T + (41.6 - 78.6i)T^{2} \) |
| 97 | \( 1 + (6.66 - 7.84i)T + (-15.6 - 95.7i)T^{2} \) |
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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
−12.15914113032807023222095798289, −11.00045616380964139404907003382, −10.16823737327764916006870030669, −9.184400663799162312222890239975, −8.388654226485846213356150181447, −6.97398350156602818049594249629, −6.09588935511958226996024874181, −4.19299422367162363639256497065, −2.55457219933069094305625203119, −0.45061487244802625853200831712,
3.14674951115333436533173614909, 4.15299679468664364483052773994, 6.11966673953544174067016032163, 7.00030769655589182835870812152, 8.386841798704374358214345076533, 9.167529710940031054899663672565, 9.733366902506931680256558141664, 11.07062162980113832199662431741, 12.26651208265468696795557665007, 13.14773138211623593500218441370
