| L(s) = 1 | + (−1.39 − 0.212i)2-s + (−0.755 − 0.654i)3-s + (1.90 + 0.595i)4-s + (3.50 + 0.504i)5-s + (0.917 + 1.07i)6-s + (1.17 + 2.56i)7-s + (−2.54 − 1.23i)8-s + (0.142 + 0.989i)9-s + (−4.79 − 1.45i)10-s + (−3.27 + 0.961i)11-s + (−1.05 − 1.70i)12-s + (−1.07 + 2.34i)13-s + (−1.09 − 3.83i)14-s + (−2.32 − 2.67i)15-s + (3.29 + 2.27i)16-s + (0.628 + 0.978i)17-s + ⋯ |
| L(s) = 1 | + (−0.988 − 0.150i)2-s + (−0.436 − 0.378i)3-s + (0.954 + 0.297i)4-s + (1.56 + 0.225i)5-s + (0.374 + 0.439i)6-s + (0.442 + 0.969i)7-s + (−0.898 − 0.438i)8-s + (0.0474 + 0.329i)9-s + (−1.51 − 0.459i)10-s + (−0.987 + 0.290i)11-s + (−0.304 − 0.490i)12-s + (−0.297 + 0.651i)13-s + (−0.291 − 1.02i)14-s + (−0.599 − 0.691i)15-s + (0.822 + 0.568i)16-s + (0.152 + 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.915605 + 0.181762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.915605 + 0.181762i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.212i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 23 | \( 1 + (1.17 + 4.64i)T \) |
| good | 5 | \( 1 + (-3.50 - 0.504i)T + (4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 2.56i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (3.27 - 0.961i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.07 - 2.34i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.628 - 0.978i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-5.54 - 3.56i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.81 + 3.73i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.69 - 3.20i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (4.31 - 0.620i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.0228 - 0.159i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-8.41 + 9.71i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 2.39iT - 47T^{2} \) |
| 53 | \( 1 + (9.60 - 4.38i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-6.32 - 2.88i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-5.96 + 5.16i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (5.38 + 1.58i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-0.0976 + 0.332i)T + (-59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (5.88 + 3.78i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.23 + 13.6i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.57 - 10.9i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.25 + 6.28i)T + (12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (14.5 + 2.08i)T + (93.0 + 27.3i)T^{2} \) |
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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
−11.98608883096888235702266177051, −10.70412225605949347760390422436, −10.08557169073162285098170066523, −9.215457701947790952202158186711, −8.205841138435067013384424787525, −7.06102530645004842679296474132, −6.00563204911673402520858828016, −5.29018453598484854979329576300, −2.60063428731887127550765640285, −1.76949456825588520179371633421,
1.13498546102226283874288192230, 2.82989769718439289478655206843, 5.11553393744099313061294725487, 5.72539077036171529839698724182, 7.03079602304392772544122210501, 7.954937034471506435458679854657, 9.300442833352252121816579456131, 9.912237832943763164578488499887, 10.61561675658696344730291400220, 11.35350583149472771242557873374
