L(s) = 1 | + (−0.564 − 1.63i)3-s + (1.59 + 2.75i)5-s + (−0.607 + 1.05i)7-s + (−2.36 + 1.84i)9-s + (−0.312 + 0.540i)11-s + (1.06 + 1.84i)13-s + (3.61 − 4.16i)15-s − 1.83·17-s − 7.15·19-s + (2.06 + 0.400i)21-s + (−0.780 − 1.35i)23-s + (−2.57 + 4.46i)25-s + (4.36 + 2.82i)27-s + (−4.87 + 8.44i)29-s + (−3.32 − 5.75i)31-s + ⋯ |
L(s) = 1 | + (−0.325 − 0.945i)3-s + (0.712 + 1.23i)5-s + (−0.229 + 0.397i)7-s + (−0.787 + 0.616i)9-s + (−0.0941 + 0.163i)11-s + (0.295 + 0.511i)13-s + (0.934 − 1.07i)15-s − 0.445·17-s − 1.64·19-s + (0.450 + 0.0874i)21-s + (−0.162 − 0.282i)23-s + (−0.515 + 0.892i)25-s + (0.839 + 0.543i)27-s + (−0.905 + 1.56i)29-s + (−0.596 − 1.03i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7848044957\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7848044957\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.564 + 1.63i)T \) |
good | 5 | \( 1 + (-1.59 - 2.75i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.607 - 1.05i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.312 - 0.540i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 1.84i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 + 7.15T + 19T^{2} \) |
| 23 | \( 1 + (0.780 + 1.35i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.87 - 8.44i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.32 + 5.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 + (5.64 + 9.77i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.51 - 7.81i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.36 + 2.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.60T + 53T^{2} \) |
| 59 | \( 1 + (-4.02 - 6.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.79 + 4.84i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.11T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 + (3.21 - 5.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.27 + 5.67i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.02T + 89T^{2} \) |
| 97 | \( 1 + (4.70 - 8.14i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.35309272270995043457990928907, −9.122368671487082071562155753385, −8.472755166914601620803566255810, −7.23028969072328662231621490694, −6.73743093233582708763113383290, −6.10587271385869591996195088158, −5.27111189107192057911236418215, −3.75621848230263906289071294349, −2.46281644344954194888477835391, −1.90339160446929825628561439847,
0.32609309655743449940316459350, 1.95795350436743480965429741904, 3.53434928752844361167789495180, 4.38522292353542842206147099054, 5.23833706720537758326660596530, 5.89006794565189735249005019692, 6.81865420115524344737063955844, 8.367135006332332509329587074601, 8.695903733410285975545961333312, 9.630347063431837308870237879254