| L(s) = 1 | + (0.377 − 1.65i)2-s + (−1.63 − 0.504i)3-s + (−0.789 − 0.380i)4-s + (0.140 + 0.358i)5-s + (−1.45 + 2.51i)6-s + (1.74 + 3.02i)7-s + (1.18 − 1.48i)8-s + (−0.0539 − 0.0367i)9-s + (0.646 − 0.0974i)10-s + (−3.90 + 1.88i)11-s + (1.10 + 1.02i)12-s + (1.26 + 0.190i)13-s + (5.65 − 1.74i)14-s + (−0.0493 − 0.658i)15-s + (−3.10 − 3.89i)16-s + (0.205 − 0.523i)17-s + ⋯ |
| L(s) = 1 | + (0.266 − 1.16i)2-s + (−0.945 − 0.291i)3-s + (−0.394 − 0.190i)4-s + (0.0629 + 0.160i)5-s + (−0.593 + 1.02i)6-s + (0.659 + 1.14i)7-s + (0.420 − 0.526i)8-s + (−0.0179 − 0.0122i)9-s + (0.204 − 0.0308i)10-s + (−1.17 + 0.567i)11-s + (0.317 + 0.294i)12-s + (0.350 + 0.0528i)13-s + (1.51 − 0.466i)14-s + (−0.0127 − 0.169i)15-s + (−0.776 − 0.974i)16-s + (0.0498 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.582637 - 0.471312i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582637 - 0.471312i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.30 + 5.66i)T \) |
| good | 2 | \( 1 + (-0.377 + 1.65i)T + (-1.80 - 0.867i)T^{2} \) |
| 3 | \( 1 + (1.63 + 0.504i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (-0.140 - 0.358i)T + (-3.66 + 3.40i)T^{2} \) |
| 7 | \( 1 + (-1.74 - 3.02i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.90 - 1.88i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (-1.26 - 0.190i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-0.205 + 0.523i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (6.30 - 4.29i)T + (6.94 - 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.553 + 7.39i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-3.26 + 1.00i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (0.717 + 0.665i)T + (2.31 + 30.9i)T^{2} \) |
| 37 | \( 1 + (2.19 - 3.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.07 - 4.72i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (-3.93 - 1.89i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (9.56 - 1.44i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (2.92 + 3.66i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-3.96 + 3.67i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-10.8 + 7.39i)T + (24.4 - 62.3i)T^{2} \) |
| 71 | \( 1 + (-0.570 - 7.61i)T + (-70.2 + 10.5i)T^{2} \) |
| 73 | \( 1 + (2.05 + 0.309i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-4.14 - 7.17i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (11.1 + 3.43i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-2.60 - 0.804i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + (-0.441 + 0.212i)T + (60.4 - 75.8i)T^{2} \) |
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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
−15.77867986154923025737622670618, −14.47355293575367265703868233382, −12.65358793519318222689180644056, −12.31238606283157845542627850133, −11.11442227747815174907090704118, −10.34052256397523903705716092709, −8.390737755265619409514004643293, −6.39495337931884295703762712286, −4.85174831086620588012152752668, −2.40151544437055866617751708950,
4.70752597874547053263193362165, 5.71227741582793477752570613285, 7.16174213584339290542154186317, 8.373492468145017320785720955154, 10.74648087961054405863360125561, 11.06388351169650205493109018912, 13.16112844759851331515275264004, 14.08225388385812521116597648117, 15.41238333851811538679735465039, 16.26250614711691837394438710937
