| L(s) = 1 | + (−0.752 + 2.24i)2-s + (−0.810 − 0.455i)3-s + (−2.88 − 2.17i)4-s + (2.05 − 3.09i)5-s + (1.63 − 1.47i)6-s + (0.245 − 0.541i)7-s + (3.15 − 2.16i)8-s + (−1.11 − 1.82i)9-s + (5.40 + 6.93i)10-s + (4.98 + 0.319i)11-s + (1.34 + 3.07i)12-s + (−0.149 − 0.172i)13-s + (1.03 + 0.958i)14-s + (−3.06 + 1.57i)15-s + (0.502 + 1.75i)16-s + (−3.36 + 1.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.531 + 1.58i)2-s + (−0.467 − 0.262i)3-s + (−1.44 − 1.08i)4-s + (0.917 − 1.38i)5-s + (0.666 − 0.603i)6-s + (0.0927 − 0.204i)7-s + (1.11 − 0.764i)8-s + (−0.370 − 0.608i)9-s + (1.71 + 2.19i)10-s + (1.50 + 0.0963i)11-s + (0.388 + 0.887i)12-s + (−0.0414 − 0.0479i)13-s + (0.275 + 0.256i)14-s + (−0.792 + 0.406i)15-s + (0.125 + 0.439i)16-s + (−0.815 + 0.360i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.640238 - 0.350012i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.640238 - 0.350012i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (-1.53 + 31.3i)T \) |
| good | 2 | \( 1 + (0.752 - 2.24i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (0.810 + 0.455i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (-2.05 + 3.09i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (-0.245 + 0.541i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (-4.98 - 0.319i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (0.149 + 0.172i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.36 - 1.48i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (5.44 + 4.15i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.224 + 2.11i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (5.50 - 7.97i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (0.0372 + 0.0554i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (-4.65 + 1.15i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (7.73 + 0.694i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (7.40 + 10.1i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (1.59 + 5.30i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-6.69 + 6.14i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.0620 - 0.00477i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (1.76 + 1.49i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (7.97 - 6.85i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (4.28 - 3.49i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-0.154 - 0.110i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (1.81 - 0.269i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (-10.2 + 3.18i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (-14.3 - 9.44i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (-7.71 + 0.395i)T + (96.4 - 9.91i)T^{2} \) |
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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
−9.231046589818603146726832735714, −8.905397135411585117508056233470, −8.496109584911371648718025124106, −6.93363638551832264847582491509, −6.61582337379002654149169048483, −5.76804412725065142477603102938, −5.04386286168710475177102987124, −4.10809324296271722555146131797, −1.68595720103543127009465802899, −0.42431388057386572189787206055,
1.76341540110645323029477807681, 2.42694496815576398927630130036, 3.53419163097476010759266300664, 4.48974884392558836695040527862, 6.02589822567315512357966095597, 6.46494475002468618517824679911, 7.87811631156871003404564800755, 8.953717738620824958064890629381, 9.631881964771986774848513142012, 10.29938848843861250719511015143
