| L(s) = 1 | + (−0.781 + 0.623i)2-s + (1.31 − 0.515i)3-s + (0.222 − 0.974i)4-s + (−2.23 − 0.0174i)5-s + (−0.705 + 1.22i)6-s + (−2.14 + 1.24i)7-s + (0.433 + 0.900i)8-s + (−0.741 + 0.687i)9-s + (1.75 − 1.38i)10-s + (0.259 + 1.13i)11-s + (−0.210 − 1.39i)12-s + (−0.426 + 0.625i)13-s + (0.906 − 2.30i)14-s + (−2.94 + 1.12i)15-s + (−0.900 − 0.433i)16-s + (−6.18 + 0.463i)17-s + ⋯ |
| L(s) = 1 | + (−0.552 + 0.440i)2-s + (0.757 − 0.297i)3-s + (0.111 − 0.487i)4-s + (−0.999 − 0.00779i)5-s + (−0.287 + 0.498i)6-s + (−0.811 + 0.468i)7-s + (0.153 + 0.318i)8-s + (−0.247 + 0.229i)9-s + (0.556 − 0.436i)10-s + (0.0783 + 0.343i)11-s + (−0.0606 − 0.402i)12-s + (−0.118 + 0.173i)13-s + (0.242 − 0.617i)14-s + (−0.760 + 0.291i)15-s + (−0.225 − 0.108i)16-s + (−1.50 + 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0739775 + 0.384848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0739775 + 0.384848i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.781 - 0.623i)T \) |
| 5 | \( 1 + (2.23 + 0.0174i)T \) |
| 43 | \( 1 + (-6.53 - 0.491i)T \) |
| good | 3 | \( 1 + (-1.31 + 0.515i)T + (2.19 - 2.04i)T^{2} \) |
| 7 | \( 1 + (2.14 - 1.24i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.259 - 1.13i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.426 - 0.625i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (6.18 - 0.463i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (0.738 + 0.684i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.05 - 3.40i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (0.858 - 2.18i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (1.05 - 0.159i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (2.11 + 1.22i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.468 + 0.587i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (2.11 + 0.482i)T + (42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.0246 + 0.0361i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (5.18 + 2.49i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (3.37 + 0.508i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-5.32 + 5.74i)T + (-5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (10.4 - 3.22i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-8.18 + 12.0i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (1.71 + 2.96i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.25 + 2.84i)T + (60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-2.26 - 5.77i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-9.90 + 2.26i)T + (87.3 - 42.0i)T^{2} \) |
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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
−11.43413228233279778271428782463, −10.64401333022000300554767424972, −9.261343294455081962241464063467, −8.891949365146252362252424398829, −7.897926456477238264992237116628, −7.17431004624762631738021041708, −6.22537988931085843158006080595, −4.79576699777198484332564247781, −3.43293898287391601347673078467, −2.18165022399716822687287310255,
0.25444752465465703498127619206, 2.61032766303633920944066874700, 3.58024233073275229152544170318, 4.37148658381926645803069847838, 6.32160944416484310571584483216, 7.27179736910124073116094365355, 8.324113363152439938887873756431, 8.883629758078894037495469760080, 9.776845406553016629741069812254, 10.73980781381698377455656996363
