| 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
−10.15297864463488020623711237017, −9.240173958746427998183195945906, −8.374569774429731489727101582656, −7.48696402712518381338186070056, −6.49134111858656258824151219843, −5.46920644587379344716949703404, −4.60037130897752101470455265877, −3.76607439246851096921693174908, −2.80278126046322018167676202600, −1.37822240561950109175096542524,
1.36811028999256549693367881719, 2.99483223685148500314395210699, 3.33784774437787370538027867417, 5.04282541590345169615301878706, 5.52608711203675994695721015630, 6.64086275735067648459575570932, 7.57515904729253600142986223201, 8.019497752880941132799990995061, 9.124654771501926362587921065551, 10.04299609741006897678370318034
