L(s) = 1 | + (−1.79 + 1.79i)2-s − 4.42i·4-s + (−0.646 + 2.14i)5-s + (−0.345 − 0.345i)7-s + (4.34 + 4.34i)8-s + (−2.67 − 4.99i)10-s + 0.0944i·11-s + (−0.829 + 0.829i)13-s + 1.23·14-s − 6.71·16-s + (−1.32 + 1.32i)17-s − 0.316i·19-s + (9.46 + 2.86i)20-s + (−0.169 − 0.169i)22-s + (0.707 + 0.707i)23-s + ⋯ |
L(s) = 1 | + (−1.26 + 1.26i)2-s − 2.21i·4-s + (−0.289 + 0.957i)5-s + (−0.130 − 0.130i)7-s + (1.53 + 1.53i)8-s + (−0.846 − 1.57i)10-s + 0.0284i·11-s + (−0.230 + 0.230i)13-s + 0.330·14-s − 1.67·16-s + (−0.322 + 0.322i)17-s − 0.0726i·19-s + (2.11 + 0.639i)20-s + (−0.0360 − 0.0360i)22-s + (0.147 + 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.04086966241\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04086966241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.646 - 2.14i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (1.79 - 1.79i)T - 2iT^{2} \) |
| 7 | \( 1 + (0.345 + 0.345i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.0944iT - 11T^{2} \) |
| 13 | \( 1 + (0.829 - 0.829i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.32 - 1.32i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.316iT - 19T^{2} \) |
| 29 | \( 1 + 4.62T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 + (-6.84 - 6.84i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.77iT - 41T^{2} \) |
| 43 | \( 1 + (1.95 - 1.95i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.30 - 4.30i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.54 + 2.54i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.28T + 59T^{2} \) |
| 61 | \( 1 + 2.16T + 61T^{2} \) |
| 67 | \( 1 + (5.18 + 5.18i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.33iT - 71T^{2} \) |
| 73 | \( 1 + (-5.77 + 5.77i)T - 73iT^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + (4.23 + 4.23i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 + (-3.36 - 3.36i)T + 97iT^{2} \) |
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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
−9.544316421353776281374649124592, −8.942722328200333577966837269014, −7.85303339090220379760043015561, −7.44848153181231430407292163879, −6.57067689616045878595378397031, −6.02885965365901286973218355183, −4.84159404461916161225005630701, −3.44721009289392429656009292865, −1.87772344764075453332551568381, −0.03024339044139772653881247049,
1.25958009446414397107265833192, 2.41164686874779330752223495144, 3.54585095648773441153356308971, 4.52458807291886284130018868748, 5.73208363558032542862147158717, 7.23679328942494886891203797974, 7.942099373525488003116685007527, 8.696671366130303265941788953689, 9.388371941395183407345983624372, 9.843506810785501358704926781914