| L(s) = 1 | + (−0.218 − 2.42i)2-s + (−3.88 + 0.704i)4-s + (−1.42 + 1.95i)5-s + (1.02 + 1.70i)7-s + (1.26 + 4.56i)8-s + (5.06 + 3.02i)10-s + (2.15 − 3.26i)11-s + (1.67 − 1.10i)13-s + (3.92 − 2.85i)14-s + (3.43 − 1.28i)16-s + (1.37 − 4.24i)17-s + (2.94 + 3.08i)19-s + (4.14 − 8.60i)20-s + (−8.40 − 4.52i)22-s + (−1.12 − 4.93i)23-s + ⋯ |
| L(s) = 1 | + (−0.154 − 1.71i)2-s + (−1.94 + 0.352i)4-s + (−0.636 + 0.875i)5-s + (0.385 + 0.645i)7-s + (0.445 + 1.61i)8-s + (1.60 + 0.957i)10-s + (0.650 − 0.984i)11-s + (0.465 − 0.307i)13-s + (1.04 − 0.762i)14-s + (0.858 − 0.322i)16-s + (0.334 − 1.02i)17-s + (0.675 + 0.706i)19-s + (0.926 − 1.92i)20-s + (−1.79 − 0.963i)22-s + (−0.234 − 1.02i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.597392 - 1.01611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.597392 - 1.01611i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (3.62 - 7.60i)T \) |
| good | 2 | \( 1 + (0.218 + 2.42i)T + (-1.96 + 0.357i)T^{2} \) |
| 5 | \( 1 + (1.42 - 1.95i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-1.02 - 1.70i)T + (-3.31 + 6.16i)T^{2} \) |
| 11 | \( 1 + (-2.15 + 3.26i)T + (-4.32 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 1.10i)T + (5.10 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-1.37 + 4.24i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.94 - 3.08i)T + (-0.852 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.12 + 4.93i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-6.29 + 3.38i)T + (15.9 - 24.2i)T^{2} \) |
| 31 | \( 1 + (-1.53 + 4.08i)T + (-23.3 - 20.3i)T^{2} \) |
| 37 | \( 1 + (1.30 - 5.69i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (3.70 - 4.65i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.52 + 5.69i)T + (5.77 - 42.6i)T^{2} \) |
| 47 | \( 1 + (0.477 + 3.52i)T + (-45.3 + 12.5i)T^{2} \) |
| 53 | \( 1 + (-8.92 - 1.61i)T + (49.6 + 18.6i)T^{2} \) |
| 59 | \( 1 + (-0.112 - 2.49i)T + (-58.7 + 5.28i)T^{2} \) |
| 61 | \( 1 + (-4.04 + 6.77i)T + (-28.9 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-1.19 - 6.57i)T + (-62.7 + 23.5i)T^{2} \) |
| 73 | \( 1 + (-1.17 + 0.105i)T + (71.8 - 13.0i)T^{2} \) |
| 79 | \( 1 + (-12.9 + 3.57i)T + (67.8 - 40.5i)T^{2} \) |
| 83 | \( 1 + (8.26 - 0.371i)T + (82.6 - 7.44i)T^{2} \) |
| 89 | \( 1 + (0.0309 - 0.170i)T + (-83.3 - 31.2i)T^{2} \) |
| 97 | \( 1 + (12.6 - 10.0i)T + (21.5 - 94.5i)T^{2} \) |
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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
−10.46781423682191422324483516498, −9.750156214243579420003071831347, −8.684214359520605106943851718348, −8.111116020243784118572773236905, −6.73727586260359159496114895307, −5.49794262060781936244119333455, −4.16834537719299093399150999107, −3.27977978040953880621045887562, −2.53080342531439189879100774482, −0.898545982534329537044455886328,
1.17367434356953322551112006956, 3.93265278102938635319296093524, 4.58519049891853774440591140503, 5.46997751768122011660059632496, 6.63653853159257099035605113957, 7.34135384169506884924183362533, 8.076056949343462191553473644141, 8.821792594102015670630221607032, 9.532257364006943136666505427001, 10.68645688188370218438383905743
