L(s) = 1 | + (0.866 − 0.5i)4-s + (0.5 − 0.866i)5-s + (1 − i)7-s + (−0.866 + 0.5i)9-s + (0.499 − 0.866i)16-s + (0.366 + 1.36i)17-s − 0.999i·20-s + (−0.366 + 1.36i)23-s + (−0.499 − 0.866i)25-s + (0.366 − 1.36i)28-s + (−0.366 − 1.36i)35-s + (−0.499 + 0.866i)36-s + (−1.36 + 0.366i)43-s + 0.999i·45-s + (−1.36 − 0.366i)47-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)4-s + (0.5 − 0.866i)5-s + (1 − i)7-s + (−0.866 + 0.5i)9-s + (0.499 − 0.866i)16-s + (0.366 + 1.36i)17-s − 0.999i·20-s + (−0.366 + 1.36i)23-s + (−0.499 − 0.866i)25-s + (0.366 − 1.36i)28-s + (−0.366 − 1.36i)35-s + (−0.499 + 0.866i)36-s + (−1.36 + 0.366i)43-s + 0.999i·45-s + (−1.36 − 0.366i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.576027531\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.576027531\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (-1 + i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1 - i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−9.464295387604048702981351662705, −8.189525313236404502595647420099, −8.055058707743443384007346282164, −6.96829873985163941699099799107, −5.95066718668048136819665674217, −5.39844900641203368948527819803, −4.58288894182124848905408330761, −3.40894672731197826436305030439, −1.95018773192803137022730556684, −1.36290705284242985829644560132,
1.90555224068989314331050168099, 2.69024534473089360619066240179, 3.30777871749459673314312383854, 4.83523607428323493651210226279, 5.73675234918728121096817945139, 6.39494738313924838814316052162, 7.14768771365253788352363500517, 8.065194557822983273295764477917, 8.653472023517116876042850829491, 9.554227985570855487638865675287