| L(s) = 1 | + (−0.576 + 1.29i)2-s + (−0.0615 + 0.0329i)3-s + (−1.33 − 1.48i)4-s + (−0.634 − 0.773i)5-s + (−0.00703 − 0.0984i)6-s + (−1.25 + 0.840i)7-s + (2.69 − 0.868i)8-s + (−1.66 + 2.49i)9-s + (1.36 − 0.374i)10-s + (5.08 − 1.54i)11-s + (0.131 + 0.0476i)12-s + (−0.464 − 0.381i)13-s + (−0.360 − 2.10i)14-s + (0.0644 + 0.0267i)15-s + (−0.429 + 3.97i)16-s + (4.92 − 2.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.407 + 0.913i)2-s + (−0.0355 + 0.0189i)3-s + (−0.668 − 0.744i)4-s + (−0.283 − 0.345i)5-s + (−0.00287 − 0.0401i)6-s + (−0.475 + 0.317i)7-s + (0.951 − 0.307i)8-s + (−0.554 + 0.830i)9-s + (0.431 − 0.118i)10-s + (1.53 − 0.465i)11-s + (0.0378 + 0.0137i)12-s + (−0.128 − 0.105i)13-s + (−0.0964 − 0.563i)14-s + (0.0166 + 0.00689i)15-s + (−0.107 + 0.994i)16-s + (1.19 − 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.528877 + 0.757562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528877 + 0.757562i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.576 - 1.29i)T \) |
| 5 | \( 1 + (0.634 + 0.773i)T \) |
| good | 3 | \( 1 + (0.0615 - 0.0329i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (1.25 - 0.840i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-5.08 + 1.54i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (0.464 + 0.381i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.92 + 2.03i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.335 - 3.40i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (6.71 - 1.33i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (2.79 - 9.20i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-3.49 - 3.49i)T + 31iT^{2} \) |
| 37 | \( 1 + (-4.78 - 0.470i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.69 - 8.54i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.87 - 2.07i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-0.799 - 1.93i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.06 - 6.81i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-4.23 + 3.47i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-3.86 - 7.22i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (5.62 + 10.5i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (2.63 + 3.94i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (2.36 + 1.58i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-4.06 + 9.81i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (2.19 - 0.216i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-12.8 - 2.56i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (2.94 + 2.94i)T + 97iT^{2} \) |
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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
−10.64309131643838239138793071633, −9.676540420049631059880979630047, −9.038138894843003752408924608578, −8.144847848850039472109384936354, −7.48518590580864939307135370610, −6.24286180322277997819831907878, −5.68178848252751385050889419342, −4.56923315941620449225462428224, −3.35282729663768515342199500699, −1.32317568664883574791808151297,
0.67011154526261339996312250518, 2.33776437555880851728639703059, 3.72471115470541375832558761808, 4.08085385418873502397282572940, 5.87941142808576087897954880816, 6.83877454074542097364338821474, 7.82534693309532850988781776808, 8.801999582448200547573194600974, 9.697715022680382143898823541187, 10.07310730711232884763725773855
