L(s) = 1 | + (0.5 − 0.866i)4-s + (1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 25-s + (−1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s − 1.73i·52-s + (1.5 − 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + (1.5 + 0.866i)97-s + (0.5 − 0.866i)100-s − 1.73i·103-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 25-s + (−1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s − 1.73i·52-s + (1.5 − 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + (1.5 + 0.866i)97-s + (0.5 − 0.866i)100-s − 1.73i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.535993797\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535993797\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.488633838646607235813003965186, −7.80075252924838030355791552784, −6.84981517408924346700867204470, −6.29974580381200253090656770581, −5.56371046908188918456572519489, −4.96477954992769936883061203332, −3.79591428457002125219018750350, −3.00137710005122716160463668300, −1.88536883880065982094522310803, −0.929211114693954615932929221806,
1.46707539226908886210842240472, 2.40538823173229163531696321226, 3.55601717595189937183127828890, 3.86665799230948920283244248292, 5.01392596827841682014063889009, 5.93867407403977119652919655640, 6.77366855644605466750994712019, 7.15261515896159011067601935442, 8.093556139875731936879771478104, 8.873187281100141177641773149947