| L(s) = 1 | + (−0.427 + 0.587i)2-s + (0.427 − 1.31i)3-s + (1.07 + 3.30i)4-s + (−3.23 + 2.35i)5-s + (0.590 + 0.812i)6-s + (−9.47 + 3.07i)7-s + (−5.16 − 1.67i)8-s + (5.73 + 4.16i)9-s − 2.90i·10-s + 4.79·12-s + (−3.09 + 4.25i)13-s + (2.23 − 6.88i)14-s + (1.70 + 5.25i)15-s + (−8.04 + 5.84i)16-s + (8.98 + 12.3i)17-s + (−4.89 + 1.59i)18-s + ⋯ |
| L(s) = 1 | + (−0.213 + 0.293i)2-s + (0.142 − 0.438i)3-s + (0.268 + 0.825i)4-s + (−0.647 + 0.470i)5-s + (0.0983 + 0.135i)6-s + (−1.35 + 0.439i)7-s + (−0.645 − 0.209i)8-s + (0.637 + 0.463i)9-s − 0.290i·10-s + 0.399·12-s + (−0.237 + 0.327i)13-s + (0.159 − 0.491i)14-s + (0.113 + 0.350i)15-s + (−0.502 + 0.365i)16-s + (0.528 + 0.727i)17-s + (−0.272 + 0.0884i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.402042 + 0.787311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.402042 + 0.787311i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (0.427 - 0.587i)T + (-1.23 - 3.80i)T^{2} \) |
| 3 | \( 1 + (-0.427 + 1.31i)T + (-7.28 - 5.29i)T^{2} \) |
| 5 | \( 1 + (3.23 - 2.35i)T + (7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (9.47 - 3.07i)T + (39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (3.09 - 4.25i)T + (-52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (-8.98 - 12.3i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (1.97 + 0.640i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + 2.76T + 529T^{2} \) |
| 29 | \( 1 + (-27.0 + 8.78i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-5.76 - 4.18i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (12.4 + 38.2i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-66.7 - 21.6i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 - 23.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-8.41 + 25.9i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-9.27 - 6.73i)T + (868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (0.701 + 2.15i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-13.2 - 18.2i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + 38.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + (61.7 - 44.8i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (97.8 - 31.7i)T + (4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (2.31 - 3.18i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-48.9 - 67.3i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 - 123.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-62.6 - 45.4i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.26387562934079955520908236775, −12.58434421638716108193661629188, −11.80349812664994238441939727418, −10.41407693393956389687931856425, −9.184789548605788092740373017568, −7.944606971524589579712828617370, −7.14249042371610457151986807588, −6.21614437076134693288182099085, −3.93516622724319978312162726576, −2.69523500502429923869748613965,
0.64950904606321192147100921711, 3.14753695351329470030971253400, 4.56956348661983277579028025895, 6.16155245118864690957709633919, 7.29211476554499500971556342399, 8.955985661442973651088564123290, 9.873602674716660109937485654865, 10.42674993848352184101205401511, 11.88186284778747151810386276860, 12.65245669547544326621554324856
