L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.0749 − 0.279i)3-s + (0.866 + 0.499i)4-s + (2.20 + 0.378i)5-s + 0.289i·6-s + (0.126 − 2.64i)7-s + (−0.707 − 0.707i)8-s + (2.52 − 1.45i)9-s + (−2.03 − 0.935i)10-s + (−2.81 + 4.87i)11-s + (0.0749 − 0.279i)12-s + (−1.42 + 1.42i)13-s + (−0.806 + 2.51i)14-s + (−0.0593 − 0.645i)15-s + (0.500 + 0.866i)16-s + (−5.12 + 1.37i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.0432 − 0.161i)3-s + (0.433 + 0.249i)4-s + (0.985 + 0.169i)5-s + 0.118i·6-s + (0.0477 − 0.998i)7-s + (−0.249 − 0.249i)8-s + (0.841 − 0.486i)9-s + (−0.642 − 0.295i)10-s + (−0.848 + 1.46i)11-s + (0.0216 − 0.0807i)12-s + (−0.396 + 0.396i)13-s + (−0.215 + 0.673i)14-s + (−0.0153 − 0.166i)15-s + (0.125 + 0.216i)16-s + (−1.24 + 0.333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.749983 - 0.145341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.749983 - 0.145341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (-2.20 - 0.378i)T \) |
| 7 | \( 1 + (-0.126 + 2.64i)T \) |
good | 3 | \( 1 + (0.0749 + 0.279i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (2.81 - 4.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.42 - 1.42i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.12 - 1.37i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.94 + 3.37i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.290 - 1.08i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3.15iT - 29T^{2} \) |
| 31 | \( 1 + (3.33 + 1.92i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.86 - 1.30i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 7.21iT - 41T^{2} \) |
| 43 | \( 1 + (-1.85 - 1.85i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.52 - 5.69i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.33 + 0.357i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.73 + 4.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.99 - 2.30i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.218 - 0.816i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + (-1.45 - 5.42i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.41 + 3.12i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.67 + 5.67i)T - 83iT^{2} \) |
| 89 | \( 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.63 + 6.63i)T + 97iT^{2} \) |
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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
−14.74407078192859634935083839242, −13.28358530152748847508894511634, −12.66507191768093066052650969418, −10.94282235670674311587353388422, −10.07674203200132231646674290396, −9.254570735152117448214152338250, −7.40220021994503878612552538465, −6.67935135699068995937512245708, −4.53043958240600486729400809543, −2.03768544672533025692614727603,
2.37046207890397897291589424561, 5.21275965683295963987468921767, 6.28711416929015966766111266866, 8.046431387517954042476418666583, 9.066030974980261201395000014415, 10.18168507749680496442110351101, 11.11920427675390542523581665800, 12.72214001808167408513283281430, 13.63042959747958888750704832307, 15.03069789881621589569830772183