L(s) = 1 | + (−0.965 − 0.258i)3-s + (−0.866 + 0.5i)5-s + (−0.448 − 0.258i)7-s + (0.866 + 0.499i)9-s + (0.965 − 0.258i)15-s + (0.366 + 0.366i)21-s + (−0.965 − 1.67i)23-s + (0.499 − 0.866i)25-s + (−0.707 − 0.707i)27-s + (−1.5 − 0.866i)29-s + 0.517·35-s + (−0.866 + 0.5i)41-s + (−1.22 − 0.707i)43-s − 45-s + (−0.258 + 0.448i)47-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)3-s + (−0.866 + 0.5i)5-s + (−0.448 − 0.258i)7-s + (0.866 + 0.499i)9-s + (0.965 − 0.258i)15-s + (0.366 + 0.366i)21-s + (−0.965 − 1.67i)23-s + (0.499 − 0.866i)25-s + (−0.707 − 0.707i)27-s + (−1.5 − 0.866i)29-s + 0.517·35-s + (−0.866 + 0.5i)41-s + (−1.22 − 0.707i)43-s − 45-s + (−0.258 + 0.448i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2805212174\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2805212174\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
good | 7 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 1.73iT - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.751709745695674660663121476299, −8.371970207523718961440225552077, −7.76736082374090352869745512760, −6.76288608311257375707677794031, −6.44187013324613609528103651165, −5.29949263772993666650714749121, −4.31920098885940584586342952165, −3.53623442669649551367976186878, −2.11789376020234597892572942799, −0.26154232795915200742277493823,
1.53206816580416030233487271724, 3.41204353638198350752088867993, 4.05747414233073226335486455276, 5.17445589569387801727767898966, 5.69225061230749362033300669737, 6.80294100256554311813908810392, 7.48094423349994771003374672092, 8.424262350645299606426200579350, 9.374766388343669659786699163268, 9.937462700670612827070873464263