| L(s) = 1 | + (0.959 + 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.0682 + 0.474i)5-s + (0.841 − 0.540i)6-s + (0.415 + 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.199 + 0.436i)10-s + (−1.38 + 0.405i)11-s + (0.959 − 0.281i)12-s + (1.74 − 3.81i)13-s + (0.142 + 0.989i)14-s + (0.314 + 0.362i)15-s + (0.415 + 0.909i)16-s + (5.58 − 3.59i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (0.378 − 0.436i)3-s + (0.420 + 0.270i)4-s + (−0.0305 + 0.212i)5-s + (0.343 − 0.220i)6-s + (0.157 + 0.343i)7-s + (0.231 + 0.267i)8-s + (−0.0474 − 0.329i)9-s + (−0.0630 + 0.137i)10-s + (−0.416 + 0.122i)11-s + (0.276 − 0.0813i)12-s + (0.483 − 1.05i)13-s + (0.0380 + 0.264i)14-s + (0.0811 + 0.0936i)15-s + (0.103 + 0.227i)16-s + (1.35 − 0.870i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.86925 + 0.0901152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.86925 + 0.0901152i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-3.74 - 2.99i)T \) |
| good | 5 | \( 1 + (0.0682 - 0.474i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (1.38 - 0.405i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 3.81i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.58 + 3.59i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.64 - 1.70i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.55 - 2.92i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.79 - 3.22i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.896 + 6.23i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.17 - 8.19i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.48 - 2.86i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 + (5.47 + 11.9i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (1.86 - 4.07i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (3.78 + 4.37i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.40 + 0.412i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.85 - 0.838i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (6.40 + 4.11i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.19 + 6.99i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.709 + 4.93i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (8.00 - 9.24i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.647 - 4.50i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.04299609741006897678370318034, −9.124654771501926362587921065551, −8.019497752880941132799990995061, −7.57515904729253600142986223201, −6.64086275735067648459575570932, −5.52608711203675994695721015630, −5.04282541590345169615301878706, −3.33784774437787370538027867417, −2.99483223685148500314395210699, −1.36811028999256549693367881719,
1.37822240561950109175096542524, 2.80278126046322018167676202600, 3.76607439246851096921693174908, 4.60037130897752101470455265877, 5.46920644587379344716949703404, 6.49134111858656258824151219843, 7.48696402712518381338186070056, 8.374569774429731489727101582656, 9.240173958746427998183195945906, 10.15297864463488020623711237017
