L(s) = 1 | − 0.236·3-s + (1.55 − 1.61i)5-s + i·7-s − 2.94·9-s + 3.36i·11-s − 1.01·13-s + (−0.366 + 0.380i)15-s + 0.988i·17-s + 2.29i·19-s − 0.236i·21-s − 0.806i·23-s + (−0.187 − 4.99i)25-s + 1.40·27-s + 3.17i·29-s + 1.77·31-s + ⋯ |
L(s) = 1 | − 0.136·3-s + (0.693 − 0.720i)5-s + 0.377i·7-s − 0.981·9-s + 1.01i·11-s − 0.281·13-s + (−0.0946 + 0.0982i)15-s + 0.239i·17-s + 0.526i·19-s − 0.0515i·21-s − 0.168i·23-s + (−0.0374 − 0.999i)25-s + 0.270·27-s + 0.589i·29-s + 0.317·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8066358152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8066358152\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.55 + 1.61i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 0.236T + 3T^{2} \) |
| 11 | \( 1 - 3.36iT - 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 - 0.988iT - 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 + 0.806iT - 23T^{2} \) |
| 29 | \( 1 - 3.17iT - 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 9.11T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 9.87iT - 47T^{2} \) |
| 53 | \( 1 + 4.36T + 53T^{2} \) |
| 59 | \( 1 - 2.34iT - 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 - 6.35T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 13.0iT - 73T^{2} \) |
| 79 | \( 1 - 7.51T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 0.147T + 89T^{2} \) |
| 97 | \( 1 - 9.75iT - 97T^{2} \) |
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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
−9.231519104054121480682275266409, −8.595999413946003078381414895966, −7.945239911603455622448361536588, −6.80862487453465090041837276176, −6.12677347960018554604855430228, −5.19868114382778780509235533107, −4.83181191375597480442385684887, −3.50941168117787393765606624769, −2.38579808187256167114697813720, −1.50341997932227205344299810509,
0.26416342935313920099984678867, 1.89588200972275652488642511362, 2.98685213965485814514147732211, 3.56874979419116426482063113338, 5.06067537251182465517134184126, 5.56623837681578289523333904338, 6.56432675629719082153582796830, 6.94184481629472860752812593103, 8.184005159615061244667339400242, 8.653265009161763210590412323011