L(s) = 1 | + (0.0966 + 0.297i)2-s + (−1.44 − 1.04i)3-s + (0.729 − 0.530i)4-s + (−0.309 + 0.951i)5-s + (0.172 − 0.530i)6-s + (0.481 + 0.349i)8-s + (0.672 + 2.06i)9-s − 0.312·10-s + (−0.951 + 0.309i)11-s − 1.60·12-s + (0.610 + 1.87i)13-s + (1.44 − 1.04i)15-s + (0.221 − 0.680i)16-s + (−0.550 + 0.400i)18-s + (0.809 + 0.587i)19-s + (0.278 + 0.857i)20-s + ⋯ |
L(s) = 1 | + (0.0966 + 0.297i)2-s + (−1.44 − 1.04i)3-s + (0.729 − 0.530i)4-s + (−0.309 + 0.951i)5-s + (0.172 − 0.530i)6-s + (0.481 + 0.349i)8-s + (0.672 + 2.06i)9-s − 0.312·10-s + (−0.951 + 0.309i)11-s − 1.60·12-s + (0.610 + 1.87i)13-s + (1.44 − 1.04i)15-s + (0.221 − 0.680i)16-s + (−0.550 + 0.400i)18-s + (0.809 + 0.587i)19-s + (0.278 + 0.857i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7375205957\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7375205957\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.951 - 0.309i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.610 - 1.87i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.437 - 1.34i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - 0.907T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.65542476454483898268835327365, −9.673299755069104978606101227983, −8.034002820146800907149286370065, −7.19651142471171418179539945298, −6.90730714384291636260161384033, −6.04978187364051797597068320119, −5.49617670156035972805251495085, −4.29623414056635024986549847591, −2.49537051034919396804455674667, −1.51047484091680220904798594506,
0.850581854587085936778647363934, 3.01678485971927833662483759855, 3.88248815049785075782786717084, 5.00917109128894673950886024309, 5.48104639492449301787903165828, 6.38189278490990326118742080308, 7.66720376459383121030968188264, 8.293036846541710386981559902991, 9.532760976411732242530638449783, 10.32475944482042833180746257003