L(s) = 1 | + (0.305 − 1.38i)2-s + (−1.81 − 0.844i)4-s + (1.45 + 1.45i)5-s + 2.37i·7-s + (−1.72 + 2.24i)8-s + (2.45 − 1.56i)10-s + (3.09 + 3.09i)11-s + (3.80 − 3.80i)13-s + (3.28 + 0.727i)14-s + (2.57 + 3.06i)16-s − 0.668·17-s + (2.02 − 2.02i)19-s + (−1.40 − 3.86i)20-s + (5.22 − 3.32i)22-s + 4.17i·23-s + ⋯ |
L(s) = 1 | + (0.216 − 0.976i)2-s + (−0.906 − 0.422i)4-s + (0.650 + 0.650i)5-s + 0.898i·7-s + (−0.608 + 0.793i)8-s + (0.776 − 0.494i)10-s + (0.933 + 0.933i)11-s + (1.05 − 1.05i)13-s + (0.877 + 0.194i)14-s + (0.643 + 0.765i)16-s − 0.162·17-s + (0.464 − 0.464i)19-s + (−0.315 − 0.864i)20-s + (1.11 − 0.709i)22-s + 0.869i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58787 - 0.387819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58787 - 0.387819i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.305 + 1.38i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.45 - 1.45i)T + 5iT^{2} \) |
| 7 | \( 1 - 2.37iT - 7T^{2} \) |
| 11 | \( 1 + (-3.09 - 3.09i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3.80 + 3.80i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.668T + 17T^{2} \) |
| 19 | \( 1 + (-2.02 + 2.02i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.17iT - 23T^{2} \) |
| 29 | \( 1 + (5.21 - 5.21i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.437T + 31T^{2} \) |
| 37 | \( 1 + (7.02 + 7.02i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.69iT - 41T^{2} \) |
| 43 | \( 1 + (-4.50 - 4.50i)T + 43iT^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 + (-5.33 - 5.33i)T + 53iT^{2} \) |
| 59 | \( 1 + (10.0 + 10.0i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.47 - 3.47i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.26 - 7.26i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.51iT - 71T^{2} \) |
| 73 | \( 1 + 15.2iT - 73T^{2} \) |
| 79 | \( 1 + 9.00T + 79T^{2} \) |
| 83 | \( 1 + (3.40 - 3.40i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.38iT - 89T^{2} \) |
| 97 | \( 1 + 1.45T + 97T^{2} \) |
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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
−10.99008416335393760589617983076, −10.39043346773415260483936843802, −9.289778373205943397797992521584, −8.891301967532602892014982334084, −7.36216376546413751687044066149, −6.03186099283831485327263705508, −5.37768085913855391579010812198, −3.90287511140427008776551193037, −2.80461689518320605856497361391, −1.64111426809420509323987585198,
1.19834447667884066140718790914, 3.65270398833177978499841277908, 4.42016675199095381010898087728, 5.74914568481226445775353050593, 6.40316467833783030047983510394, 7.39341952356490594825216494652, 8.638911456173647980028761175281, 9.050234700933904490703993520759, 10.08934053676727898788867837761, 11.28732604498724785163509086432