L(s) = 1 | + (0.766 + 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.5 + 0.866i)6-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)10-s + (−0.173 + 0.984i)12-s + (0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (1.17 − 0.984i)17-s + (0.173 + 0.984i)18-s + (−0.939 − 0.342i)19-s + (0.939 − 0.342i)20-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.5 + 0.866i)6-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)10-s + (−0.173 + 0.984i)12-s + (0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (1.17 − 0.984i)17-s + (0.173 + 0.984i)18-s + (−0.939 − 0.342i)19-s + (0.939 − 0.342i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.993661251\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.993661251\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.939 + 0.342i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (0.173 - 0.984i)T^{2} \) |
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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
−9.903353160205854882127223565860, −9.057340836812768842734986168685, −8.399107360909927704550696182959, −7.76927583888768010915687116296, −6.95213489593413373671065407989, −5.69335787028668368085101131035, −4.90672646519154371466290220787, −4.12109554881202569840669920438, −3.28419831848148057081633483950, −2.02717899314325414056212562150,
1.76912538127038890589413262383, 2.59875610616561907544341286418, 3.73418233580871279849124378344, 4.07075213493097948639690634871, 5.82990904200008689052880223832, 6.31543966679764293949008885236, 7.48037265007152680828686555504, 8.050705645009583009071320217031, 9.280846039724308875585188317831, 10.07796530491913350374885293939