| L(s) = 1 | + (1.38 − 0.287i)2-s + (1.31 + 1.64i)3-s + (1.83 − 0.795i)4-s + (−2.15 + 1.71i)5-s + (2.28 + 1.90i)6-s + (−0.0757 − 2.64i)7-s + (2.31 − 1.62i)8-s + (−0.317 + 1.38i)9-s + (−2.48 + 2.99i)10-s + (−5.04 + 1.15i)11-s + (3.71 + 1.97i)12-s + (4.41 − 1.00i)13-s + (−0.864 − 3.64i)14-s + (−5.64 − 1.28i)15-s + (2.73 − 2.91i)16-s + (−1.55 + 3.23i)17-s + ⋯ |
| L(s) = 1 | + (0.979 − 0.203i)2-s + (0.757 + 0.949i)3-s + (0.917 − 0.397i)4-s + (−0.962 + 0.767i)5-s + (0.934 + 0.775i)6-s + (−0.0286 − 0.999i)7-s + (0.817 − 0.575i)8-s + (−0.105 + 0.463i)9-s + (−0.786 + 0.946i)10-s + (−1.52 + 0.347i)11-s + (1.07 + 0.570i)12-s + (1.22 − 0.279i)13-s + (−0.231 − 0.972i)14-s + (−1.45 − 0.332i)15-s + (0.683 − 0.729i)16-s + (−0.377 + 0.784i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12479 + 0.468665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12479 + 0.468665i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 0.287i)T \) |
| 7 | \( 1 + (0.0757 + 2.64i)T \) |
| good | 3 | \( 1 + (-1.31 - 1.64i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (2.15 - 1.71i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (5.04 - 1.15i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-4.41 + 1.00i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (1.55 - 3.23i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 + (3.00 + 6.24i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (2.62 + 1.26i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + 2.18T + 31T^{2} \) |
| 37 | \( 1 + (-9.60 - 4.62i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (2.45 - 1.95i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (0.804 + 0.641i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-0.583 - 2.55i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-3.85 + 1.85i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (1.51 - 1.89i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (5.08 - 10.5i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 - 13.2iT - 67T^{2} \) |
| 71 | \( 1 + (-1.49 - 3.11i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (8.97 + 2.04i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + 4.00iT - 79T^{2} \) |
| 83 | \( 1 + (-0.261 + 1.14i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-12.0 - 2.76i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 - 2.97iT - 97T^{2} \) |
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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
−12.86185839250867776243073515528, −11.38420528862572209574488290706, −10.52934852933545762662806913607, −10.23783946503593984143939958364, −8.345878045951090455916595388838, −7.48477219489677528835765266982, −6.22452993246934717594738415941, −4.45878624473715993772287825580, −3.83683576308199453596817950109, −2.81234615364351205431316384111,
2.14671551897986856252501679973, 3.43799245975935930841323189450, 4.94788556245965909363779751676, 6.04942203323068915735135049436, 7.54818403133363915328290555878, 8.071672591050218683219055196361, 8.942266202205957047475819263724, 10.99703248067377532726731016131, 11.79834764411887281983610897999, 12.76915601800572486065884861173
