L(s) = 1 | + (−0.760 + 1.19i)2-s + (−0.844 − 1.81i)4-s + (−0.432 − 2.19i)5-s + (0.611 − 0.611i)7-s + (2.80 + 0.371i)8-s + (2.94 + 1.15i)10-s − 5.12i·11-s + (1.76 − 1.76i)13-s + (0.264 + 1.19i)14-s + (−2.57 + 3.06i)16-s + (3.76 + 3.76i)17-s − 1.22·19-s + (−3.61 + 2.63i)20-s + (6.11 + 3.89i)22-s + (1.07 + 1.07i)23-s + ⋯ |
L(s) = 1 | + (−0.537 + 0.843i)2-s + (−0.422 − 0.906i)4-s + (−0.193 − 0.981i)5-s + (0.231 − 0.231i)7-s + (0.991 + 0.131i)8-s + (0.931 + 0.364i)10-s − 1.54i·11-s + (0.488 − 0.488i)13-s + (0.0706 + 0.319i)14-s + (−0.643 + 0.765i)16-s + (0.912 + 0.912i)17-s − 0.280·19-s + (−0.807 + 0.589i)20-s + (1.30 + 0.831i)22-s + (0.224 + 0.224i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.823990 - 0.161418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.823990 - 0.161418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.760 - 1.19i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.432 + 2.19i)T \) |
good | 7 | \( 1 + (-0.611 + 0.611i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.12iT - 11T^{2} \) |
| 13 | \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.76 - 3.76i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 + (-1.07 - 1.07i)T + 23iT^{2} \) |
| 29 | \( 1 - 0.864iT - 29T^{2} \) |
| 31 | \( 1 + 7.81iT - 31T^{2} \) |
| 37 | \( 1 + (1.76 + 1.76i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.52T + 41T^{2} \) |
| 43 | \( 1 + (-6.20 - 6.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.29 - 2.29i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.62 + 2.62i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.528T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 + (6.20 - 6.20i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.10iT - 71T^{2} \) |
| 73 | \( 1 + (2.25 - 2.25i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + (-7.95 - 7.95i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.25iT - 89T^{2} \) |
| 97 | \( 1 + (-0.793 - 0.793i)T + 97iT^{2} \) |
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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
−12.87119386846386195058196916451, −11.42067099961631596489304979428, −10.50657087949719459855957238042, −9.291271623697500058576021414204, −8.343338511771553417849264973941, −7.82805609193348548699691867961, −6.14217224269777977690468233329, −5.38492835423364979382298714587, −3.90379381599095151135882938982, −1.02283280735355297591449413594,
2.05496452210949819973067628220, 3.44372213168457646151327199460, 4.83600937250297009824545157794, 6.82622201429034966282020355317, 7.61201332568599515439802672023, 8.888840691983440821701484147568, 9.957077293746591732797186153377, 10.63891990675676500669108652617, 11.77940947167556988234181809038, 12.27917003807651866545763115120