| L(s) = 1 | + (−0.187 − 0.884i)2-s + (−1.04 + 3.22i)3-s + (1.08 − 0.480i)4-s + (0.0125 − 0.119i)5-s + (3.04 + 0.320i)6-s + (2.28 + 2.05i)7-s + (−1.69 − 2.32i)8-s + (−6.86 − 4.98i)9-s + (−0.108 + 0.0113i)10-s − 2.01i·11-s + (0.418 + 3.98i)12-s + (−1.36 − 2.36i)13-s + (1.38 − 2.40i)14-s + (0.372 + 0.165i)15-s + (−0.158 + 0.175i)16-s + (0.906 + 2.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.132 − 0.625i)2-s + (−0.604 + 1.86i)3-s + (0.540 − 0.240i)4-s + (0.00562 − 0.0535i)5-s + (1.24 + 0.130i)6-s + (0.863 + 0.777i)7-s + (−0.597 − 0.823i)8-s + (−2.28 − 1.66i)9-s + (−0.0342 + 0.00359i)10-s − 0.608i·11-s + (0.120 + 1.15i)12-s + (−0.379 − 0.656i)13-s + (0.371 − 0.643i)14-s + (0.0961 + 0.0428i)15-s + (−0.0395 + 0.0439i)16-s + (0.219 + 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.781040 + 0.188259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.781040 + 0.188259i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (-1.77 + 7.60i)T \) |
| good | 2 | \( 1 + (0.187 + 0.884i)T + (-1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (1.04 - 3.22i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.0125 + 0.119i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-2.28 - 2.05i)T + (0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + 2.01iT - 11T^{2} \) |
| 13 | \( 1 + (1.36 + 2.36i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.906 - 2.03i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (1.28 + 1.43i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.488 + 0.672i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (5.16 + 2.98i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.968 - 4.55i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (4.81 - 1.56i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.57 - 4.86i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (2.80 - 6.30i)T + (-28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (3.36 - 5.83i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.93 - 2.66i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.814 - 3.83i)T + (-53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (4.66 + 0.490i)T + (65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-9.17 + 0.964i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (1.03 + 9.81i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.32 + 2.97i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.296 - 0.0629i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (11.1 + 3.61i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (4.88 + 1.03i)T + (88.6 + 39.4i)T^{2} \) |
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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
−15.13662158199743110007226549320, −14.70301631604025025609855770311, −12.35237813254362291239983461827, −11.29452036204813904344613212309, −10.78650349158584865457203918173, −9.713827078903861547032140340188, −8.600558483192515532968410684831, −6.01718853306286375108454839445, −4.95016685730422483048982513621, −3.16783308469294545535136525484,
1.98569684138852291662715174652, 5.40521724930637185107053376399, 6.94508877297862913166867021192, 7.29361891634376751271514375354, 8.418159671727396341761753741107, 10.91682306220236617579117074073, 11.76014336716481850736992805824, 12.63623237982066488370436670989, 13.92426131745714829355759101327, 14.73903933151761426480449134340
