| L(s) = 1 | + (1.04 − 0.957i)2-s + (1.57 + 1.96i)3-s + (0.168 − 1.99i)4-s + (1.55 − 1.23i)5-s + (3.52 + 0.547i)6-s + (−2.03 + 1.69i)7-s + (−1.73 − 2.23i)8-s + (−0.745 + 3.26i)9-s + (0.432 − 2.77i)10-s + (−3.09 + 0.707i)11-s + (4.18 − 2.79i)12-s + (−5.38 + 1.22i)13-s + (−0.498 + 3.70i)14-s + (4.88 + 1.11i)15-s + (−3.94 − 0.670i)16-s + (2.48 − 5.15i)17-s + ⋯ |
| L(s) = 1 | + (0.736 − 0.676i)2-s + (0.906 + 1.13i)3-s + (0.0840 − 0.996i)4-s + (0.695 − 0.554i)5-s + (1.43 + 0.223i)6-s + (−0.768 + 0.639i)7-s + (−0.612 − 0.790i)8-s + (−0.248 + 1.08i)9-s + (0.136 − 0.878i)10-s + (−0.934 + 0.213i)11-s + (1.20 − 0.808i)12-s + (−1.49 + 0.340i)13-s + (−0.133 + 0.991i)14-s + (1.26 + 0.287i)15-s + (−0.985 − 0.167i)16-s + (0.601 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08034 - 0.313948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08034 - 0.313948i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.04 + 0.957i)T \) |
| 7 | \( 1 + (2.03 - 1.69i)T \) |
| good | 3 | \( 1 + (-1.57 - 1.96i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (-1.55 + 1.23i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (3.09 - 0.707i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (5.38 - 1.22i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 5.15i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 - 7.59T + 19T^{2} \) |
| 23 | \( 1 + (-0.751 - 1.55i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-0.626 - 0.301i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + 8.05T + 31T^{2} \) |
| 37 | \( 1 + (-4.24 - 2.04i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (0.0970 - 0.0773i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (3.53 + 2.81i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-0.299 - 1.31i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.183 + 0.0884i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-1.40 + 1.76i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-0.524 + 1.08i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 - 3.71iT - 67T^{2} \) |
| 71 | \( 1 + (-3.42 - 7.11i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.61 - 1.05i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + 9.99iT - 79T^{2} \) |
| 83 | \( 1 + (0.712 - 3.12i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-14.8 - 3.38i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + 3.30iT - 97T^{2} \) |
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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
−12.55440896627499980084445453130, −11.61938895117308715832859662879, −10.07777061883088520295446107280, −9.603468683642423225285906882156, −9.196425649424016647575140029358, −7.33898344789991881400230652559, −5.39030496015750653749336225199, −5.00068880229603840637837373321, −3.32786904440539284213164332162, −2.47853955719616050727838115526,
2.46337090368210571077251427719, 3.36711194796252915514281565424, 5.36499230183182803013202877550, 6.50298114340818633096466249985, 7.46100841187498582417516901444, 7.899389600321459782877685197492, 9.439752795377984949325272864340, 10.49881660107492147250323224062, 12.20314931366054305651439906705, 12.92287510094218929709998181684
