L(s) = 1 | + (0.0915 − 1.41i)2-s + (1.49 − 2.59i)3-s + (−1.98 − 0.258i)4-s + (−0.866 + 0.5i)5-s + (−3.52 − 2.35i)6-s + (2.06 + 1.65i)7-s + (−0.546 + 2.77i)8-s + (−2.99 − 5.18i)9-s + (0.626 + 1.26i)10-s + (1.93 + 1.11i)11-s + (−3.64 + 4.76i)12-s + 3.17i·13-s + (2.52 − 2.75i)14-s + 2.99i·15-s + (3.86 + 1.02i)16-s + (−2.98 − 1.72i)17-s + ⋯ |
L(s) = 1 | + (0.0647 − 0.997i)2-s + (0.865 − 1.49i)3-s + (−0.991 − 0.129i)4-s + (−0.387 + 0.223i)5-s + (−1.43 − 0.960i)6-s + (0.778 + 0.627i)7-s + (−0.193 + 0.981i)8-s + (−0.998 − 1.72i)9-s + (0.198 + 0.400i)10-s + (0.584 + 0.337i)11-s + (−1.05 + 1.37i)12-s + 0.879i·13-s + (0.676 − 0.736i)14-s + 0.774i·15-s + (0.966 + 0.256i)16-s + (−0.723 − 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.560978 - 1.17598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.560978 - 1.17598i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0915 + 1.41i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-2.06 - 1.65i)T \) |
good | 3 | \( 1 + (-1.49 + 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.93 - 1.11i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.17iT - 13T^{2} \) |
| 17 | \( 1 + (2.98 + 1.72i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.02 + 1.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.30 + 1.33i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 + (-2.44 + 4.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.59 - 9.69i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.95iT - 43T^{2} \) |
| 47 | \( 1 + (3.06 + 5.30i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.32 - 4.03i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.19 - 1.26i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0456 - 0.0263i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.212iT - 71T^{2} \) |
| 73 | \( 1 + (12.8 + 7.43i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.399 - 0.230i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + (-6.07 + 3.51i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.185iT - 97T^{2} \) |
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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
−12.82592186489966781436999236792, −11.68213889577964285256706086044, −11.40860102122929142696215116019, −9.414848825582525488123294553849, −8.656475991023175789593617662327, −7.69546574053836092130561842528, −6.43990826407911910927452873431, −4.48245514255955596853407628478, −2.76655547946686439723719666365, −1.66126434518535741943017728961,
3.63037610013973777395881448394, 4.38632761921007374570550890095, 5.53348027256752660465602480540, 7.42609238751831084620351594360, 8.409796115399264563671751933981, 9.065090904929816847781054423310, 10.23507606179341772294873097208, 11.15228878360838373288713932403, 12.92971597215409315762441681752, 14.00354466009090905813929813779