L(s) = 1 | + 1.27i·2-s + (0.583 − 1.01i)3-s + 0.370·4-s + (1.57 + 0.907i)5-s + (1.29 + 0.745i)6-s + 3.02i·8-s + (0.817 + 1.41i)9-s + (−1.15 + 2.00i)10-s + (2.40 + 1.38i)11-s + (0.216 − 0.374i)12-s + (−3.58 − 0.402i)13-s + (1.83 − 1.05i)15-s − 3.12·16-s − 2.74·17-s + (−1.80 + 1.04i)18-s + (5.08 − 2.93i)19-s + ⋯ |
L(s) = 1 | + 0.902i·2-s + (0.337 − 0.583i)3-s + 0.185·4-s + (0.702 + 0.405i)5-s + (0.527 + 0.304i)6-s + 1.06i·8-s + (0.272 + 0.472i)9-s + (−0.366 + 0.634i)10-s + (0.725 + 0.418i)11-s + (0.0624 − 0.108i)12-s + (−0.993 − 0.111i)13-s + (0.473 − 0.273i)15-s − 0.780·16-s − 0.665·17-s + (−0.426 + 0.246i)18-s + (1.16 − 0.673i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69177 + 1.31225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69177 + 1.31225i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.58 + 0.402i)T \) |
good | 2 | \( 1 - 1.27iT - 2T^{2} \) |
| 3 | \( 1 + (-0.583 + 1.01i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.57 - 0.907i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.40 - 1.38i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 2.74T + 17T^{2} \) |
| 19 | \( 1 + (-5.08 + 2.93i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.79 + 1.03i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.74iT - 37T^{2} \) |
| 41 | \( 1 + (-5.51 + 3.18i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.55 + 7.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.76 + 3.32i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.24 - 9.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 3.07iT - 59T^{2} \) |
| 61 | \( 1 + (0.540 + 0.936i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.34 + 2.50i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.35 - 1.35i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.64 - 3.83i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.86 + 13.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.97iT - 83T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 + (12.3 + 7.11i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.62173759456994348991830253365, −9.827545296614182891342317037824, −8.841850498345377555644473114532, −7.77636028971408627880900554623, −7.18787094301381599276220904040, −6.53516315848215209705724632447, −5.57987109611810271576829282620, −4.52194926579488812877100355296, −2.65184053021041713941785789926, −1.90199270524595445907021938291,
1.26382935065783460890187967880, 2.48947337808136539436399652491, 3.62648909105687383994464634637, 4.47206905984375505757952837862, 5.85569094247099518857479155605, 6.71849890661787393301478674957, 7.897241878085504760715317001944, 9.213986117414690032896706938286, 9.675098893603315224495017730940, 10.14758500809978161935369400094