L(s) = 1 | + (−0.334 + 0.334i)5-s + 4.55i·7-s + (2.47 − 2.47i)11-s + (0.0594 + 0.0594i)13-s − 3.61·17-s + (2.55 + 2.55i)19-s + 2.82i·23-s + 4.77i·25-s + (−5.16 − 5.16i)29-s + 0.557·31-s + (−1.52 − 1.52i)35-s + (−4.38 + 4.38i)37-s + 9.27i·41-s + (−1.61 + 1.61i)43-s + 2.82·47-s + ⋯ |
L(s) = 1 | + (−0.149 + 0.149i)5-s + 1.72i·7-s + (0.745 − 0.745i)11-s + (0.0164 + 0.0164i)13-s − 0.877·17-s + (0.586 + 0.586i)19-s + 0.589i·23-s + 0.955i·25-s + (−0.958 − 0.958i)29-s + 0.100·31-s + (−0.258 − 0.258i)35-s + (−0.721 + 0.721i)37-s + 1.44i·41-s + (−0.245 + 0.245i)43-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.291478859\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.291478859\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.334 - 0.334i)T - 5iT^{2} \) |
| 7 | \( 1 - 4.55iT - 7T^{2} \) |
| 11 | \( 1 + (-2.47 + 2.47i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.0594 - 0.0594i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + (-2.55 - 2.55i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (5.16 + 5.16i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 + (4.38 - 4.38i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.27iT - 41T^{2} \) |
| 43 | \( 1 + (1.61 - 1.61i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (0.493 - 0.493i)T - 53iT^{2} \) |
| 59 | \( 1 + (4 - 4i)T - 59iT^{2} \) |
| 61 | \( 1 + (2.72 + 2.72i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.77 - 3.77i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.11iT - 71T^{2} \) |
| 73 | \( 1 + 0.541iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−9.772357509111399676319486338056, −9.189002486734100139141242819448, −8.531272168960200608656199974212, −7.69383452353418717657722151502, −6.50514133034329217093258936248, −5.86227678497890758329708703794, −5.07607058069834984791505567065, −3.75732279203597202039562600613, −2.83147404710934290085600512635, −1.64854184242485776290998916874,
0.56703357513918046513125123159, 1.94517964183321982380805126794, 3.55644915261042393834848244146, 4.26087299736389811290514798841, 5.03616756785209964365924449069, 6.51021005101125704788205357453, 7.06286471870003836577271209837, 7.71747136170142585733181392906, 8.860356379965984476641709628317, 9.520858490780498706321182083417