L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s − 0.618i·7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)12-s + (−1.53 − 0.5i)13-s + (0.309 − 0.951i)16-s + (−1.30 − 0.951i)19-s + (−0.500 + 0.363i)21-s + (0.951 − 0.309i)27-s + (−0.363 − 0.500i)28-s + (−0.5 − 0.363i)31-s + (0.309 + 0.951i)36-s + (1.53 + 0.5i)37-s + (0.5 + 1.53i)39-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s − 0.618i·7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)12-s + (−1.53 − 0.5i)13-s + (0.309 − 0.951i)16-s + (−1.30 − 0.951i)19-s + (−0.500 + 0.363i)21-s + (0.951 − 0.309i)27-s + (−0.363 − 0.500i)28-s + (−0.5 − 0.363i)31-s + (0.309 + 0.951i)36-s + (1.53 + 0.5i)37-s + (0.5 + 1.53i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8724578844\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8724578844\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + 0.618iT - T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61iT - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−9.179179907889797845582783276836, −8.010341240765013054078438677910, −7.29058487049344590977901972852, −6.87037596278704256721679313288, −6.01634414422805126638399369363, −5.23487814431435460871853198594, −4.39290315835317192497091038834, −2.72426124109663175909985096038, −2.05552584792015777628946839849, −0.63600572203016021202852322439,
2.03264511494882419310764728661, 2.92714604720364587821509943433, 4.03037443363655278021942662940, 4.77060131338814090211665352559, 5.84678147873480905653810767034, 6.40893606755074180314765910856, 7.32575388279465442912136215439, 8.155539003980007661363403983387, 9.053317250867078605586395841552, 9.768231297903781715944238338177