| L(s) = 1 | + (1.40 − 0.174i)2-s + (0.428 − 1.38i)3-s + (1.93 − 0.490i)4-s + (−1.89 − 2.04i)5-s + (0.358 − 2.02i)6-s + (2.04 + 1.68i)7-s + (2.63 − 1.02i)8-s + (0.735 + 0.501i)9-s + (−3.01 − 2.53i)10-s + (−1.25 − 1.84i)11-s + (0.149 − 2.90i)12-s + (−1.37 + 2.85i)13-s + (3.16 + 2.00i)14-s + (−3.64 + 1.75i)15-s + (3.51 − 1.90i)16-s + (−1.80 − 4.60i)17-s + ⋯ |
| L(s) = 1 | + (0.992 − 0.123i)2-s + (0.247 − 0.801i)3-s + (0.969 − 0.245i)4-s + (−0.847 − 0.913i)5-s + (0.146 − 0.825i)6-s + (0.772 + 0.635i)7-s + (0.931 − 0.362i)8-s + (0.245 + 0.167i)9-s + (−0.954 − 0.802i)10-s + (−0.379 − 0.555i)11-s + (0.0432 − 0.837i)12-s + (−0.381 + 0.792i)13-s + (0.844 + 0.535i)14-s + (−0.941 + 0.453i)15-s + (0.879 − 0.475i)16-s + (−0.438 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04953 - 1.46421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04953 - 1.46421i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.174i)T \) |
| 7 | \( 1 + (-2.04 - 1.68i)T \) |
| good | 3 | \( 1 + (-0.428 + 1.38i)T + (-2.47 - 1.68i)T^{2} \) |
| 5 | \( 1 + (1.89 + 2.04i)T + (-0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (1.25 + 1.84i)T + (-4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (1.37 - 2.85i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (1.80 + 4.60i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (3.05 - 1.76i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.243 - 0.620i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-1.79 - 1.43i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (2.45 - 4.24i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.04 - 6.92i)T + (-35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-2.32 - 10.1i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-4.45 - 1.01i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.0584 - 0.779i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-1.63 - 10.8i)T + (-50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-8.20 + 8.84i)T + (-4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (0.970 - 6.44i)T + (-58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (4.99 + 2.88i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.3 + 12.9i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.749 + 10.0i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (4.74 + 8.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.11 + 4.38i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-8.79 - 5.99i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−11.64184056364862070498203593287, −10.59371323267199513558307924420, −9.038705408193888650235636493329, −8.092784515127810046059584355464, −7.42588364068713530726864236690, −6.30471021818449809980693293043, −4.93960855357262759762551705810, −4.44534985685243222037793194415, −2.74681318926200326025941712377, −1.48496312420053715504513290698,
2.41408856785899819371366640253, 3.86927798018962106287490866463, 4.18837211534917442440836327912, 5.43950595184968368700603588410, 6.91347901784481001727675961030, 7.48384464509281473975610091305, 8.478604096256562147806811155387, 10.19177910152220286599612449408, 10.66328122937434119824108809761, 11.32980502234223756104978090334
