| L(s) = 1 | + (−0.980 − 0.330i)2-s + (1.02 + 1.39i)3-s + (−0.740 − 0.563i)4-s + (−2.53 − 1.01i)5-s + (−0.544 − 1.70i)6-s + (−3.45 + 1.59i)7-s + (1.70 + 2.50i)8-s + (−0.894 + 2.86i)9-s + (2.15 + 1.82i)10-s + (−0.332 + 1.19i)11-s + (0.0258 − 1.61i)12-s + (−3.53 − 1.87i)13-s + (3.91 − 0.425i)14-s + (−1.19 − 4.57i)15-s + (−0.340 − 1.22i)16-s + (−0.947 + 2.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.693 − 0.233i)2-s + (0.592 + 0.805i)3-s + (−0.370 − 0.281i)4-s + (−1.13 − 0.452i)5-s + (−0.222 − 0.696i)6-s + (−1.30 + 0.603i)7-s + (0.601 + 0.886i)8-s + (−0.298 + 0.954i)9-s + (0.680 + 0.578i)10-s + (−0.100 + 0.360i)11-s + (0.00746 − 0.465i)12-s + (−0.979 − 0.519i)13-s + (1.04 − 0.113i)14-s + (−0.307 − 1.18i)15-s + (−0.0851 − 0.306i)16-s + (−0.229 + 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0469090 + 0.206402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0469090 + 0.206402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.02 - 1.39i)T \) |
| 59 | \( 1 + (-5.46 - 5.39i)T \) |
| good | 2 | \( 1 + (0.980 + 0.330i)T + (1.59 + 1.21i)T^{2} \) |
| 5 | \( 1 + (2.53 + 1.01i)T + (3.62 + 3.43i)T^{2} \) |
| 7 | \( 1 + (3.45 - 1.59i)T + (4.53 - 5.33i)T^{2} \) |
| 11 | \( 1 + (0.332 - 1.19i)T + (-9.42 - 5.67i)T^{2} \) |
| 13 | \( 1 + (3.53 + 1.87i)T + (7.29 + 10.7i)T^{2} \) |
| 17 | \( 1 + (0.947 - 2.04i)T + (-11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (-0.558 + 0.122i)T + (17.2 - 7.97i)T^{2} \) |
| 23 | \( 1 + (-1.16 + 7.08i)T + (-21.7 - 7.34i)T^{2} \) |
| 29 | \( 1 + (-3.01 - 8.95i)T + (-23.0 + 17.5i)T^{2} \) |
| 31 | \( 1 + (1.30 - 5.90i)T + (-28.1 - 13.0i)T^{2} \) |
| 37 | \( 1 + (4.73 + 3.20i)T + (13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (0.441 - 0.0722i)T + (38.8 - 13.0i)T^{2} \) |
| 43 | \( 1 + (-10.2 + 2.84i)T + (36.8 - 22.1i)T^{2} \) |
| 47 | \( 1 + (1.85 + 4.65i)T + (-34.1 + 32.3i)T^{2} \) |
| 53 | \( 1 + (2.67 - 2.27i)T + (8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (2.25 - 6.68i)T + (-48.5 - 36.9i)T^{2} \) |
| 67 | \( 1 + (6.37 - 4.31i)T + (24.7 - 62.2i)T^{2} \) |
| 71 | \( 1 + (11.7 - 4.66i)T + (51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (-1.07 - 9.88i)T + (-71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (-3.59 + 2.16i)T + (37.0 - 69.7i)T^{2} \) |
| 83 | \( 1 + (0.413 + 7.62i)T + (-82.5 + 8.97i)T^{2} \) |
| 89 | \( 1 + (11.0 - 3.71i)T + (70.8 - 53.8i)T^{2} \) |
| 97 | \( 1 + (-0.349 + 3.21i)T + (-94.7 - 20.8i)T^{2} \) |
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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
−12.86772024337402607327776966491, −12.21520826230789325591286077056, −10.66111304474146421593834190619, −10.09454140561506380520698775777, −8.943773714812624405402019191631, −8.582570816583780302800003381488, −7.27047960213836687791600070087, −5.32391916605690956685277917915, −4.26165982920513847036036718959, −2.81116048879661752542199954159,
0.22032527823003849476333487894, 3.10395486866799783625230093914, 4.07204771228283752438128762880, 6.51210602243820503336548468320, 7.44448889745602433333582275303, 7.82831842119447234725865046834, 9.245719496833975614696377909513, 9.819244763835341985397450857755, 11.42168146772946505589708187670, 12.34867870891571146180972125864
