| L(s) = 1 | + (0.734 − 1.20i)2-s + (0.0712 + 0.0760i)3-s + (−0.922 − 1.77i)4-s + (−1.69 − 3.74i)5-s + (0.144 − 0.0302i)6-s + (2.17 − 1.50i)7-s + (−2.82 − 0.187i)8-s + (0.195 − 2.98i)9-s + (−5.77 − 0.697i)10-s + (−1.01 − 1.63i)11-s + (0.0692 − 0.196i)12-s + (0.353 + 3.58i)13-s + (−0.226 − 3.73i)14-s + (0.163 − 0.395i)15-s + (−2.29 + 3.27i)16-s + (5.93 + 0.781i)17-s + ⋯ |
| L(s) = 1 | + (0.519 − 0.854i)2-s + (0.0411 + 0.0439i)3-s + (−0.461 − 0.887i)4-s + (−0.758 − 1.67i)5-s + (0.0589 − 0.0123i)6-s + (0.821 − 0.569i)7-s + (−0.997 − 0.0663i)8-s + (0.0651 − 0.994i)9-s + (−1.82 − 0.220i)10-s + (−0.306 − 0.493i)11-s + (0.0200 − 0.0567i)12-s + (0.0980 + 0.995i)13-s + (−0.0606 − 0.998i)14-s + (0.0423 − 0.102i)15-s + (−0.574 + 0.818i)16-s + (1.43 + 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.238918 + 1.64194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.238918 + 1.64194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.734 + 1.20i)T \) |
| 7 | \( 1 + (-2.17 + 1.50i)T \) |
| good | 3 | \( 1 + (-0.0712 - 0.0760i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.69 + 3.74i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (1.01 + 1.63i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (-0.353 - 3.58i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-5.93 - 0.781i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (0.173 - 1.05i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (1.05 + 1.20i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.87 - 7.24i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (0.769 - 0.206i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-2.44 + 6.48i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (1.38 + 6.93i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (10.9 - 3.30i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-7.07 + 9.22i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-5.80 - 9.34i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (-4.48 + 6.25i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (5.78 - 1.34i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (8.38 + 8.95i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-8.27 - 12.3i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-1.67 + 0.828i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-0.0947 - 0.719i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (5.37 - 6.54i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-0.337 - 0.993i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (-4.98 + 4.98i)T - 97iT^{2} \) |
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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
−9.681874295202040508822934094083, −8.793751159497490221917425556457, −8.361552768911636021264177028777, −7.15469187200532332180991767376, −5.72649375937757734010902213218, −4.99010786221286939331332941290, −4.11086568643098571039112571739, −3.54064859432423106922111347219, −1.56603549384772497957288243382, −0.71934840979258973256156020816,
2.52193719433965457054429944558, 3.24296399613591460282836222613, 4.49279748903797140874923089252, 5.39246154595683668907141474381, 6.28673002558479830182386181017, 7.37606328804345428348601406335, 7.85925598038052242297357957391, 8.261434989223227511126866159307, 9.919865647868508476268623994496, 10.54323873240156054069053075428
