L(s) = 1 | + (1.21 − 1.21i)3-s + (−0.714 + 0.232i)5-s + (0.736 − 4.65i)7-s + 0.0368i·9-s + (1.45 + 2.85i)11-s + (0.0718 − 0.0113i)13-s + (−0.587 + 1.15i)15-s + (0.390 − 0.199i)17-s + (−3.87 − 0.613i)19-s + (−4.76 − 6.55i)21-s + (6.27 + 4.55i)23-s + (−3.58 + 2.60i)25-s + (3.69 + 3.69i)27-s + (−2.52 − 1.28i)29-s + (−0.161 + 0.498i)31-s + ⋯ |
L(s) = 1 | + (0.702 − 0.702i)3-s + (−0.319 + 0.103i)5-s + (0.278 − 1.75i)7-s + 0.0122i·9-s + (0.438 + 0.861i)11-s + (0.0199 − 0.00315i)13-s + (−0.151 + 0.297i)15-s + (0.0948 − 0.0483i)17-s + (−0.888 − 0.140i)19-s + (−1.04 − 1.43i)21-s + (1.30 + 0.950i)23-s + (−0.717 + 0.521i)25-s + (0.711 + 0.711i)27-s + (−0.469 − 0.239i)29-s + (−0.0290 + 0.0894i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23357 - 0.567414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23357 - 0.567414i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 41 | \( 1 + (6.31 - 1.06i)T \) |
good | 3 | \( 1 + (-1.21 + 1.21i)T - 3iT^{2} \) |
| 5 | \( 1 + (0.714 - 0.232i)T + (4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.736 + 4.65i)T + (-6.65 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 2.85i)T + (-6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (-0.0718 + 0.0113i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.390 + 0.199i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (3.87 + 0.613i)T + (18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (-6.27 - 4.55i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.52 + 1.28i)T + (17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (0.161 - 0.498i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.14 - 3.53i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (5.49 - 7.56i)T + (-13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (1.42 + 8.99i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-1.70 - 0.867i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-11.4 - 8.30i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.0743 - 0.102i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.60 + 11.0i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (1.14 + 2.24i)T + (-41.7 + 57.4i)T^{2} \) |
| 73 | \( 1 + 5.71iT - 73T^{2} \) |
| 79 | \( 1 + (-7.67 + 7.67i)T - 79iT^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + (-1.05 + 6.65i)T + (-84.6 - 27.5i)T^{2} \) |
| 97 | \( 1 + (-3.29 + 6.46i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−13.20076911771150632535185061995, −11.69065665923992810540665493421, −10.73206034835808749732068658287, −9.696486454145324230725298195280, −8.313971277074138109914076018921, −7.38221995286677267042878337714, −6.88185097449152079051187998173, −4.74897013461386459314810045343, −3.53545675891725904714263727079, −1.60657212503846089825303092586,
2.55146695112355483790343742017, 3.84518600625954565213714686872, 5.28670869662145650400941420306, 6.48238311942942341017904534523, 8.476339157740676142423422026370, 8.650000274544156781594301258961, 9.702417562015369567131225199605, 11.06266641465494635339225935247, 11.97340469651648063565307503172, 12.85065765615390277226682322660