L(s) = 1 | − 0.129·2-s + 15.5i·3-s − 63.9·4-s − 126. i·5-s − 2.02i·6-s + (−213. − 268. i)7-s + 16.6·8-s − 243·9-s + 16.4i·10-s − 1.71e3·11-s − 997. i·12-s − 309. i·13-s + (27.7 + 34.9i)14-s + 1.97e3·15-s + 4.09e3·16-s + 5.99e3i·17-s + ⋯ |
L(s) = 1 | − 0.0162·2-s + 0.577i·3-s − 0.999·4-s − 1.01i·5-s − 0.00937i·6-s + (−0.621 − 0.783i)7-s + 0.0324·8-s − 0.333·9-s + 0.0164i·10-s − 1.28·11-s − 0.577i·12-s − 0.140i·13-s + (0.0101 + 0.0127i)14-s + 0.584·15-s + 0.999·16-s + 1.21i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.132244 - 0.379229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.132244 - 0.379229i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5iT \) |
| 7 | \( 1 + (213. + 268. i)T \) |
good | 2 | \( 1 + 0.129T + 64T^{2} \) |
| 5 | \( 1 + 126. iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 1.71e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 309. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 5.99e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.03e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 409.T + 1.48e8T^{2} \) |
| 29 | \( 1 + 3.31e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 5.24e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.40e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.93e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 8.66e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.80e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.44e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 7.49e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.98e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.26e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 2.38e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.25e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 1.76e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.95e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.95e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 4.51e5iT - 8.32e11T^{2} \) |
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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
−16.52787453347624392779064907341, −15.14710664997439899688950197136, −13.42384145578077867038203132194, −12.80872783109216524192032029782, −10.55205056059064598271523937292, −9.399117421088725490848453201164, −8.077242417473807055458347979108, −5.37319586950891599586888990915, −3.97585723722715981618332622758, −0.25388523137312724193240707728,
2.92668868912059041952565702748, 5.51875752181301761266133738164, 7.33031644593306693440166478054, 8.998015485612958357102622854773, 10.49273207915836871673423126461, 12.29667988844002112849654121090, 13.45006817563620628108874353655, 14.56450088062672726544158380170, 15.99156203005538473367297805003, 17.84653658771273623215695481803