| L(s) = 1 | + (−0.123 − 0.100i)2-s + (1.69 + 0.376i)3-s + (−0.410 − 1.93i)4-s + (1.70 − 1.45i)5-s + (−0.171 − 0.215i)6-s + (−0.310 + 0.0831i)7-s + (−0.286 + 0.563i)8-s + (2.71 + 1.27i)9-s + (−0.355 + 0.00889i)10-s + (−4.68 + 0.492i)11-s + (0.0341 − 3.42i)12-s + (−2.00 − 2.47i)13-s + (0.0466 + 0.0207i)14-s + (3.42 − 1.80i)15-s + (−3.51 + 1.56i)16-s + (5.06 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.0874 − 0.0707i)2-s + (0.976 + 0.217i)3-s + (−0.205 − 0.965i)4-s + (0.761 − 0.648i)5-s + (−0.0699 − 0.0881i)6-s + (−0.117 + 0.0314i)7-s + (−0.101 + 0.199i)8-s + (0.905 + 0.424i)9-s + (−0.112 + 0.00281i)10-s + (−1.41 + 0.148i)11-s + (0.00984 − 0.987i)12-s + (−0.555 − 0.686i)13-s + (0.0124 + 0.00555i)14-s + (0.884 − 0.467i)15-s + (−0.879 + 0.391i)16-s + (1.22 + 0.626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47755 - 0.583181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47755 - 0.583181i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.69 - 0.376i)T \) |
| 5 | \( 1 + (-1.70 + 1.45i)T \) |
| good | 2 | \( 1 + (0.123 + 0.100i)T + (0.415 + 1.95i)T^{2} \) |
| 7 | \( 1 + (0.310 - 0.0831i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (4.68 - 0.492i)T + (10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (2.00 + 2.47i)T + (-2.70 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-5.06 - 2.58i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.57 + 0.837i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.47 - 6.44i)T + (-17.0 + 15.3i)T^{2} \) |
| 29 | \( 1 + (-4.56 - 5.07i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (5.69 - 6.32i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.909 + 5.74i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-1.09 - 0.114i)T + (40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (2.35 + 8.79i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (9.69 + 0.507i)T + (46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (0.855 - 0.435i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.200 + 1.90i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 9.92i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-5.53 + 0.290i)T + (66.6 - 7.00i)T^{2} \) |
| 71 | \( 1 + (7.70 + 2.50i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.784 + 4.95i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.42 + 1.28i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.996 - 1.53i)T + (-33.7 + 75.8i)T^{2} \) |
| 89 | \( 1 + (5.87 - 4.26i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.582 + 11.1i)T + (-96.4 - 10.1i)T^{2} \) |
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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
−12.50963644103906748949901558816, −10.63768534446646092624886722623, −10.07940386189515479465600712233, −9.362267305871249813573438843633, −8.382198405043937497647653931363, −7.29309224771679423599021416543, −5.47832868508914884008376670360, −5.06121129888169463286370060676, −3.09486271637047998368453187138, −1.59861688006700564145552633742,
2.48267541583832085243942535991, 3.22867299769838769304740226461, 4.86799757295249694690692140745, 6.57397852031992857064970372056, 7.56760803613442091709008715399, 8.216903632844662643658712393724, 9.532205236756124309780945803302, 10.02734549349019356756444895886, 11.50446991987251268049263383801, 12.68713449791672074339835980899
