| L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (0.889 − 1.02i)5-s + (0.281 − 0.959i)6-s + (−2.54 + 0.734i)7-s + (−0.415 − 0.909i)8-s + (0.654 + 0.755i)9-s + (1.14 + 0.734i)10-s + (4.24 + 0.610i)11-s + (0.989 + 0.142i)12-s + (−0.475 + 0.739i)13-s + (−1.08 − 2.41i)14-s + (−1.23 + 0.564i)15-s + (0.841 − 0.540i)16-s + (−3.28 − 0.964i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (0.397 − 0.458i)5-s + (0.115 − 0.391i)6-s + (−0.960 + 0.277i)7-s + (−0.146 − 0.321i)8-s + (0.218 + 0.251i)9-s + (0.361 + 0.232i)10-s + (1.28 + 0.184i)11-s + (0.285 + 0.0410i)12-s + (−0.131 + 0.205i)13-s + (−0.290 − 0.644i)14-s + (−0.318 + 0.145i)15-s + (0.210 − 0.135i)16-s + (−0.796 − 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23016 + 0.493122i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23016 + 0.493122i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (2.54 - 0.734i)T \) |
| 23 | \( 1 + (-1.17 + 4.65i)T \) |
| good | 5 | \( 1 + (-0.889 + 1.02i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-4.24 - 0.610i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.475 - 0.739i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.28 + 0.964i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.33 + 0.393i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-7.19 - 2.11i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-7.83 + 3.57i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.18 - 2.75i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.57 - 2.23i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-9.02 - 4.12i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 6.53iT - 47T^{2} \) |
| 53 | \( 1 + (-6.35 - 9.88i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-7.72 + 12.0i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 2.27i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (11.2 - 1.62i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.321 + 2.23i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (1.79 + 6.11i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-7.04 + 10.9i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.62 - 7.64i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.82 - 14.9i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.30 + 9.58i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.833074455019723144162833826835, −9.228692628292214465665729228077, −8.581317747074597386141118740860, −7.31790188747160234444725521921, −6.45152862360200112768889358347, −6.18219953787022651280254874011, −4.94268566546859894723556231333, −4.19776337100169261427854061212, −2.72678374825749822479538222056, −0.996196132047855682761401555788,
0.894136390172298526849793727111, 2.49825430392214350073426006183, 3.59306504511784150866874508584, 4.36916326827422702799131325262, 5.63802023280223982387526508123, 6.45284856023748059482987633644, 7.06228141234896310686217988918, 8.608451587568916835071166949413, 9.320867692974511422816937446976, 10.20876450966483094851473381443
