| L(s) = 1 | − 3.22·3-s − 4.22i·5-s + i·7-s + 7.41·9-s − 2.60i·11-s + (3.22 − 1.60i)13-s + 13.6i·15-s + 2.56·17-s − 3.60i·19-s − 3.22i·21-s + 3.61·23-s − 12.8·25-s − 14.2·27-s − 8.03·29-s − 1.56i·31-s + ⋯ |
| L(s) = 1 | − 1.86·3-s − 1.89i·5-s + 0.377i·7-s + 2.47·9-s − 0.786i·11-s + (0.895 − 0.445i)13-s + 3.52i·15-s + 0.622·17-s − 0.827i·19-s − 0.704i·21-s + 0.754·23-s − 2.57·25-s − 2.74·27-s − 1.49·29-s − 0.281i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.144137 - 0.612632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144137 - 0.612632i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-3.22 + 1.60i)T \) |
| good | 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 + 4.22iT - 5T^{2} \) |
| 11 | \( 1 + 2.60iT - 11T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 3.60iT - 19T^{2} \) |
| 23 | \( 1 - 3.61T + 23T^{2} \) |
| 29 | \( 1 + 8.03T + 29T^{2} \) |
| 31 | \( 1 + 1.56iT - 31T^{2} \) |
| 37 | \( 1 - 3.84iT - 37T^{2} \) |
| 41 | \( 1 + 3.79iT - 41T^{2} \) |
| 43 | \( 1 + 0.419T + 43T^{2} \) |
| 47 | \( 1 + 10.6iT - 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 + 0.165iT - 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 4.65iT - 67T^{2} \) |
| 71 | \( 1 - 7.64iT - 71T^{2} \) |
| 73 | \( 1 - 0.188iT - 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 2.30iT - 83T^{2} \) |
| 89 | \( 1 + 9.22iT - 89T^{2} \) |
| 97 | \( 1 - 19.0iT - 97T^{2} \) |
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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
−10.15092903773176590857419901961, −9.158489568709954014871497880848, −8.464260440662457754590894665139, −7.30102396469954169671093462013, −6.01067109770512307778148748916, −5.51971094617062513608846399375, −4.93089407597364307394795086989, −3.86648355927968578051906481274, −1.37873590570041759549903154318, −0.46850962627993191881430381618,
1.64060628642518175050684531396, 3.43791194376026823673403277593, 4.41755811186363659944553432966, 5.76285052575554788178243552633, 6.25753955601354429382875670380, 7.14791282695791093599079855834, 7.55092302896183672916571618422, 9.555086506906478718359074508888, 10.26134923358785840702410502364, 10.91273794285456712593301841100
