L(s) = 1 | + (−3.67 + 3.67i)3-s + (4.94 + 24.5i)5-s + (−52.2 − 52.2i)7-s − 27i·9-s − 102.·11-s + (−203. + 203. i)13-s + (−108. − 71.8i)15-s + (−194. − 194. i)17-s + 178. i·19-s + 383.·21-s + (456. − 456. i)23-s + (−576. + 242. i)25-s + (99.2 + 99.2i)27-s + 1.06e3i·29-s + 552.·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.197 + 0.980i)5-s + (−1.06 − 1.06i)7-s − 0.333i·9-s − 0.849·11-s + (−1.20 + 1.20i)13-s + (−0.480 − 0.319i)15-s + (−0.673 − 0.673i)17-s + 0.494i·19-s + 0.870·21-s + (0.862 − 0.862i)23-s + (−0.921 + 0.387i)25-s + (0.136 + 0.136i)27-s + 1.27i·29-s + 0.574·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0329i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.00531807 + 0.323134i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00531807 + 0.323134i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.67 - 3.67i)T \) |
| 5 | \( 1 + (-4.94 - 24.5i)T \) |
good | 7 | \( 1 + (52.2 + 52.2i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 102.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (203. - 203. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (194. + 194. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 178. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-456. + 456. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 1.06e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 552.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-223. - 223. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 3.06e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (2.30e3 - 2.30e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (964. + 964. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (1.61e3 - 1.61e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + 870. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.33e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (3.05e3 + 3.05e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 1.73e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (2.94e3 - 2.94e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 2.70e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-777. + 777. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 6.58e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (4.13e3 + 4.13e3i)T + 8.85e7iT^{2} \) |
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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
−14.82254672260832992765566674199, −13.86411125106712815210750990501, −12.69602961273402884072614590280, −11.21877667830428161826584686103, −10.29414710622388067328898689896, −9.476547730295539741063549861541, −7.27330934147482245502217390839, −6.50238938461038962088018973641, −4.59929043636936695657432808011, −2.91849018054907592396123581487,
0.17739548307531552967008497018, 2.56978428598701003118810173891, 5.06793063842064826217149550712, 6.04310913942516550112456851321, 7.75368076300301545670512655757, 9.078138715244960517114512941785, 10.17280578989574579681586979446, 11.82186766959521900497976450456, 12.88400519005733817799778398809, 13.14275208559492710697879734445