L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1 − 1.73i)5-s + (0.5 + 2.59i)7-s + 0.999·8-s + (0.999 + 1.73i)10-s + (−2.5 − 4.33i)11-s + 6·13-s + (−2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + (2 − 3.46i)19-s − 1.99·20-s + 5·22-s + (2 − 3.46i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s + (0.188 + 0.981i)7-s + 0.353·8-s + (0.316 + 0.547i)10-s + (−0.753 − 1.30i)11-s + 1.66·13-s + (−0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + (0.458 − 0.794i)19-s − 0.447·20-s + 1.06·22-s + (0.417 − 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24995 + 0.159286i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24995 + 0.159286i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7T + 29T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−11.21375243520116772446232837754, −10.48790127298936222542777956699, −9.146730707315217859400558859832, −8.616110163791181481917739112365, −8.051297276954217384620094255539, −6.35151123794455552933910014669, −5.75481172076231567331258024339, −4.86306975132392431662723791250, −3.10477536134618031274930925188, −1.22174811552632889663503935042,
1.44323726919418435034615483217, 2.94937132716818921035815316197, 4.06873918968581365007151186214, 5.41528924757612415312792822050, 6.85957519119754392339192978441, 7.52801294749679246787525157448, 8.636317006050173519329877872436, 9.930793068154725754043953358116, 10.32430256752543438701754242353, 11.08320596721790441503109658364