L(s) = 1 | + (2.35 + 0.631i)2-s + (1.54 − 0.775i)3-s + (3.42 + 1.97i)4-s + (0.0540 + 2.23i)5-s + (4.13 − 0.849i)6-s + (3.36 + 3.36i)8-s + (1.79 − 2.40i)9-s + (−1.28 + 5.30i)10-s + (−3.08 − 1.77i)11-s + (6.83 + 0.406i)12-s + (−1.28 + 1.28i)13-s + (1.81 + 3.42i)15-s + (1.85 + 3.21i)16-s + (−0.792 − 2.95i)17-s + (5.75 − 4.52i)18-s + (−0.331 + 0.191i)19-s + ⋯ |
L(s) = 1 | + (1.66 + 0.446i)2-s + (0.894 − 0.447i)3-s + (1.71 + 0.987i)4-s + (0.0241 + 0.999i)5-s + (1.68 − 0.346i)6-s + (1.19 + 1.19i)8-s + (0.599 − 0.800i)9-s + (−0.406 + 1.67i)10-s + (−0.928 − 0.536i)11-s + (1.97 + 0.117i)12-s + (−0.356 + 0.356i)13-s + (0.469 + 0.883i)15-s + (0.463 + 0.803i)16-s + (−0.192 − 0.717i)17-s + (1.35 − 1.06i)18-s + (−0.0761 + 0.0439i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.59133 + 1.53442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.59133 + 1.53442i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.54 + 0.775i)T \) |
| 5 | \( 1 + (-0.0540 - 2.23i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-2.35 - 0.631i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (3.08 + 1.77i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.28 - 1.28i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.792 + 2.95i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.331 - 0.191i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.658 - 2.45i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + (0.323 - 0.561i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.34 - 5.00i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (0.335 - 0.335i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.80 + 0.751i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.04 - 0.815i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.81 - 6.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.45 - 9.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.3 + 3.31i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 3.06iT - 71T^{2} \) |
| 73 | \( 1 + (-0.849 - 3.17i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.21 - 1.85i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.51 + 2.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 - 10.3i)T + 97iT^{2} \) |
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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
−10.67027989553052414343540106097, −9.669166551044939829252483868463, −8.386333846741498156165546396165, −7.44650343611972430188968139455, −6.92083988107910636190317726337, −6.07933222697746027199166522828, −5.02916844415141714858973146840, −3.86113869674634745987500835180, −2.99706676039841897354222883836, −2.33402017192151560249593917201,
1.87528326042320519549139945855, 2.81082443367346819022158755380, 3.92770586734981815708925979591, 4.74209753470522569694622369276, 5.24419010228761397195053235748, 6.46605356952020749402191871009, 7.78730699133631540782771616469, 8.488868988382008205293892660805, 9.685519445337878563101614119218, 10.37433494819173380184427041291