L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s − i·6-s + (0.942 − 2.47i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (1.55 − 2.69i)11-s + (−0.258 + 0.965i)12-s + (−3.40 + 3.40i)13-s + (−1.55 + 2.14i)14-s + (0.500 + 0.866i)16-s + (5.14 − 1.37i)17-s + (0.965 − 0.258i)18-s + (−3.61 − 6.26i)19-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s − 0.408i·6-s + (0.356 − 0.934i)7-s + (−0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.468 − 0.811i)11-s + (−0.0747 + 0.278i)12-s + (−0.945 + 0.945i)13-s + (−0.414 + 0.572i)14-s + (0.125 + 0.216i)16-s + (1.24 − 0.334i)17-s + (0.227 − 0.0610i)18-s + (−0.829 − 1.43i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.041588715\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041588715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.942 + 2.47i)T \) |
good | 11 | \( 1 + (-1.55 + 2.69i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.40 - 3.40i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.14 + 1.37i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.61 + 6.26i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.36 + 5.08i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 4.49iT - 29T^{2} \) |
| 31 | \( 1 + (7.98 + 4.61i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.47 + 0.929i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 2.51iT - 41T^{2} \) |
| 43 | \( 1 + (-3.86 - 3.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.05 + 3.94i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.32 + 0.890i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.666 + 1.15i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 6.27i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.72 - 6.43i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 9.22T + 71T^{2} \) |
| 73 | \( 1 + (2.13 + 7.95i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.65 + 0.957i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.97 + 8.97i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.03 - 3.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.69 - 2.69i)T + 97iT^{2} \) |
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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
−9.642155980477920737484374647878, −9.067160886360142649777658934466, −8.274558410548956221635750854468, −7.26104811556612121726905141335, −6.70337662387891602915869660662, −5.31270311419412977708275320806, −4.35098328248377496657341322146, −3.42622610864778649359830641062, −2.18265280446959545586573620392, −0.59243007856238214903129272281,
1.45705042830228541666206855385, 2.37578716949144741804854534464, 3.66510028813360302384859692621, 5.29505876757781617424259293112, 5.80014957570963740064835930403, 6.95408708498951322497102479663, 7.76779332255619866655522416484, 8.233089747852562577417183944876, 9.257976493532893999237514512397, 9.901797407285885164693899733093