| L(s) = 1 | + (−0.409 − 1.35i)2-s + (−0.446 − 1.67i)3-s + (−1.66 + 1.10i)4-s + (−0.512 + 1.40i)5-s + (−2.08 + 1.28i)6-s + (−0.984 − 0.173i)7-s + (2.18 + 1.79i)8-s + (−2.60 + 1.49i)9-s + (2.11 + 0.116i)10-s + (−0.394 + 0.143i)11-s + (2.59 + 2.29i)12-s + (−1.30 + 1.09i)13-s + (0.168 + 1.40i)14-s + (2.58 + 0.229i)15-s + (1.53 − 3.69i)16-s + (1.20 − 0.693i)17-s + ⋯ |
| L(s) = 1 | + (−0.289 − 0.957i)2-s + (−0.257 − 0.966i)3-s + (−0.832 + 0.554i)4-s + (−0.228 + 0.629i)5-s + (−0.850 + 0.526i)6-s + (−0.372 − 0.0656i)7-s + (0.771 + 0.635i)8-s + (−0.867 + 0.497i)9-s + (0.668 + 0.0368i)10-s + (−0.118 + 0.0432i)11-s + (0.750 + 0.661i)12-s + (−0.363 + 0.304i)13-s + (0.0450 + 0.375i)14-s + (0.666 + 0.0591i)15-s + (0.384 − 0.922i)16-s + (0.291 − 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.811630 - 0.254771i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.811630 - 0.254771i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.409 + 1.35i)T \) |
| 3 | \( 1 + (0.446 + 1.67i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (0.512 - 1.40i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (0.394 - 0.143i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.30 - 1.09i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.20 + 0.693i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.55 - 2.05i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.450 - 2.55i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.96 + 5.91i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.98 + 0.526i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.90 - 6.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0493 - 0.0588i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.727 - 1.99i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.389 - 2.20i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 7.46iT - 53T^{2} \) |
| 59 | \( 1 + (3.56 + 1.29i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.06 - 11.7i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.44 - 2.91i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.04 - 7.01i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.567 - 0.983i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.11 - 6.09i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.45 + 5.41i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (7.55 + 4.36i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.87 + 2.13i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.26049157530460587318326479973, −9.674460441627211172253947643368, −8.502188368504975315778449039175, −7.69468057589744190437735469126, −7.00038395732287149260725411472, −5.88828005280547988628031954271, −4.70218318401552844477719408494, −3.30344940045912615208508530075, −2.52070752657468036826248777313, −1.09278496249246774400220171132,
0.62591708243015902361077256808, 3.13827764115844155456000932004, 4.41635093641567930504663256830, 5.04975246759431862123644800203, 5.90209480033910175750161730709, 6.89035662021874830493473964837, 7.988033243981936359062747285635, 8.796430277445483944916958207904, 9.408281367257135982358650179489, 10.22671407373808484135522330300
