L(s) = 1 | + (−0.639 + 2.38i)2-s + i·3-s + (−3.54 − 2.04i)4-s + (−0.746 − 2.78i)5-s + (−2.38 − 0.639i)6-s + (−2.52 − 0.794i)7-s + (3.65 − 3.65i)8-s − 9-s + 7.11·10-s + (0.990 − 0.990i)11-s + (2.04 − 3.54i)12-s + (−3.49 − 0.872i)13-s + (3.50 − 5.51i)14-s + (2.78 − 0.746i)15-s + (2.29 + 3.96i)16-s + (3.77 − 6.54i)17-s + ⋯ |
L(s) = 1 | + (−0.451 + 1.68i)2-s + 0.577i·3-s + (−1.77 − 1.02i)4-s + (−0.333 − 1.24i)5-s + (−0.973 − 0.260i)6-s + (−0.953 − 0.300i)7-s + (1.29 − 1.29i)8-s − 0.333·9-s + 2.25·10-s + (0.298 − 0.298i)11-s + (0.591 − 1.02i)12-s + (−0.970 − 0.242i)13-s + (0.937 − 1.47i)14-s + (0.719 − 0.192i)15-s + (0.572 + 0.992i)16-s + (0.916 − 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.265893 - 0.0896677i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.265893 - 0.0896677i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (2.52 + 0.794i)T \) |
| 13 | \( 1 + (3.49 + 0.872i)T \) |
good | 2 | \( 1 + (0.639 - 2.38i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (0.746 + 2.78i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.990 + 0.990i)T - 11iT^{2} \) |
| 17 | \( 1 + (-3.77 + 6.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.88 - 4.88i)T - 19iT^{2} \) |
| 23 | \( 1 + (1.97 - 1.13i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.75 - 6.49i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.19 + 1.39i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.20 + 0.591i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.38 + 8.90i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.04 + 1.75i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.833 - 0.223i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.886 + 1.53i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.19 + 1.39i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 4.18iT - 61T^{2} \) |
| 67 | \( 1 + (3.93 + 3.93i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.75 - 6.54i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-2.17 + 8.10i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.411 + 0.713i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.15 - 2.15i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.666 - 2.48i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (8.79 + 2.35i)T + (84.0 + 48.5i)T^{2} \) |
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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
−12.05852335354360781422561198362, −10.33129415395517145935907550750, −9.431224854041561217193049105819, −8.931931094911472402481641682040, −7.86315983963681810094556215579, −7.02227202147824116016900884955, −5.69808539423042925130348995351, −5.02455587838148619433565834088, −3.78255938851167297264053075828, −0.23970096101114486740474210357,
2.09754812270105010344903113751, 3.02947256678184424229801700168, 4.08584720624295515359098948707, 6.19252753618387871633625524350, 7.20435134190701965790652031216, 8.408463791717121155694176491506, 9.540248908729507651125470619513, 10.26352186242717269780407053343, 11.05904463557487057282055643299, 11.92721533020202950623831902580