L(s) = 1 | + (1.71 − 0.243i)3-s + (1.34 + 2.32i)5-s + (−2.48 + 4.30i)7-s + (2.88 − 0.835i)9-s + (−1.26 + 2.19i)11-s + (−2.21 − 3.83i)13-s + (2.86 + 3.65i)15-s − 2.43·17-s + 4.18·19-s + (−3.21 + 7.99i)21-s + (0.570 + 0.988i)23-s + (−1.09 + 1.89i)25-s + (4.73 − 2.13i)27-s + (−3.00 + 5.20i)29-s + (2.65 + 4.59i)31-s + ⋯ |
L(s) = 1 | + (0.990 − 0.140i)3-s + (0.599 + 1.03i)5-s + (−0.939 + 1.62i)7-s + (0.960 − 0.278i)9-s + (−0.382 + 0.662i)11-s + (−0.614 − 1.06i)13-s + (0.739 + 0.943i)15-s − 0.590·17-s + 0.959·19-s + (−0.701 + 1.74i)21-s + (0.118 + 0.206i)23-s + (−0.218 + 0.378i)25-s + (0.911 − 0.410i)27-s + (−0.557 + 0.966i)29-s + (0.476 + 0.825i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.078740339\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.078740339\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 + 0.243i)T \) |
good | 5 | \( 1 + (-1.34 - 2.32i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.48 - 4.30i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.26 - 2.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.21 + 3.83i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 - 4.18T + 19T^{2} \) |
| 23 | \( 1 + (-0.570 - 0.988i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.00 - 5.20i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.65 - 4.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.241T + 37T^{2} \) |
| 41 | \( 1 + (3.21 + 5.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.57 - 9.65i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.37 + 4.11i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 + (-5.40 - 9.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.16 - 7.20i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.13 + 1.96i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.52T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 + (-3.14 + 5.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.738 - 1.27i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + (-5.89 + 10.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.03929800390940172742344959399, −9.186177816387046861839854602960, −8.573603041077547907407552906112, −7.41539159506051113764077230488, −6.84138158594749189261184307124, −5.86859367973349526998144136514, −5.02494301300010195349696262016, −3.27719281046246611122386608367, −2.82028260873861646918278258743, −2.05718503242246555835068592255,
0.77759481909765734659507511143, 2.13717814725539256076359151823, 3.43099290298258344261351372522, 4.22598462196857046463446864923, 5.06005881859727874330207407207, 6.40680210434690128836865361985, 7.19252792252977526723510838149, 7.972310484479793339959895140929, 8.932133704115250160243215772693, 9.624224219777218276605175346283