L(s) = 1 | + (−1.41 + 0.0182i)2-s + (1.99 − 0.0515i)4-s + (−0.785 + 0.453i)5-s + (−2.47 − 0.947i)7-s + (−2.82 + 0.109i)8-s + (1.10 − 0.655i)10-s + (−0.0729 + 0.126i)11-s + 6.12·13-s + (3.51 + 1.29i)14-s + (3.99 − 0.206i)16-s + (−3.00 + 5.20i)17-s + (2.10 + 3.64i)19-s + (−1.54 + 0.946i)20-s + (0.100 − 0.179i)22-s + (3.20 − 1.85i)23-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0128i)2-s + (0.999 − 0.0257i)4-s + (−0.351 + 0.202i)5-s + (−0.933 − 0.357i)7-s + (−0.999 + 0.0386i)8-s + (0.348 − 0.207i)10-s + (−0.0219 + 0.0380i)11-s + 1.69·13-s + (0.938 + 0.345i)14-s + (0.998 − 0.0515i)16-s + (−0.728 + 1.26i)17-s + (0.482 + 0.836i)19-s + (−0.345 + 0.211i)20-s + (0.0214 − 0.0383i)22-s + (0.668 − 0.385i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.665207 + 0.337248i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.665207 + 0.337248i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0182i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.47 + 0.947i)T \) |
good | 5 | \( 1 + (0.785 - 0.453i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.0729 - 0.126i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 + (3.00 - 5.20i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.10 - 3.64i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.20 + 1.85i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + (3.54 + 2.04i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.51 - 1.45i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.26T + 41T^{2} \) |
| 43 | \( 1 - 8.73iT - 43T^{2} \) |
| 47 | \( 1 + (-3.58 - 6.20i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.86 - 3.23i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.35 - 3.66i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.41 - 5.90i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.66 + 1.54i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.91iT - 71T^{2} \) |
| 73 | \( 1 + (-2.67 - 1.54i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.41 + 9.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + (-1.55 - 2.69i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.593iT - 97T^{2} \) |
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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
−10.77325569473495056046126271129, −10.28996869402495828577216500366, −9.181274962951853719674479014873, −8.468621199932185070219385405568, −7.56835557582071011232287362834, −6.48692485706963625408275890035, −5.99438506563283402533208448345, −3.97966067883975431876913147123, −3.04675287203261828213474196067, −1.27752376839958781806107619412,
0.70559522181438679436934117314, 2.58128067237532207070641299865, 3.65668239243041156367106461075, 5.34395965589514347712071656594, 6.52853153806192696864777002143, 7.07953606791411253641933190513, 8.448284532329534026385761736642, 8.897192512607936339155986566544, 9.753261903908245380203319087570, 10.74844592501334865216062789922