L(s) = 1 | − 3.31i·2-s + (−2.5 − 1.65i)3-s − 7·4-s + (−5.5 + 8.29i)6-s + 9.94i·8-s + (3.5 + 8.29i)9-s − 16.5i·11-s + (17.5 + 11.6i)12-s − 10·13-s + 5.00·16-s − 3.31i·17-s + (27.4 − 11.6i)18-s + 7·19-s − 55.0·22-s − 19.8i·23-s + (16.5 − 24.8i)24-s + ⋯ |
L(s) = 1 | − 1.65i·2-s + (−0.833 − 0.552i)3-s − 1.75·4-s + (−0.916 + 1.38i)6-s + 1.24i·8-s + (0.388 + 0.921i)9-s − 1.50i·11-s + (1.45 + 0.967i)12-s − 0.769·13-s + 0.312·16-s − 0.195i·17-s + (1.52 − 0.644i)18-s + 0.368·19-s − 2.50·22-s − 0.865i·23-s + (0.687 − 1.03i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.212652 + 0.705287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.212652 + 0.705287i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.5 + 1.65i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 3.31iT - 4T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 + 16.5iT - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 + 3.31iT - 289T^{2} \) |
| 19 | \( 1 - 7T + 361T^{2} \) |
| 23 | \( 1 + 19.8iT - 529T^{2} \) |
| 29 | \( 1 + 33.1iT - 841T^{2} \) |
| 31 | \( 1 - 42T + 961T^{2} \) |
| 37 | \( 1 - 40T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 45T + 4.48e3T^{2} \) |
| 71 | \( 1 + 33.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35T + 5.32e3T^{2} \) |
| 79 | \( 1 - 12T + 6.24e3T^{2} \) |
| 83 | \( 1 + 69.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 149. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 70T + 9.40e3T^{2} \) |
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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
−13.25897015978267317957162402064, −12.26707260963880281144351454594, −11.47019098443622841110943855599, −10.71170401709734369883665800005, −9.573970437635075759008966940938, −8.031786887265594906957224790805, −6.17892388696250017207784049152, −4.59375851392575445562101445835, −2.70426003703818706466712273760, −0.72507732653626953368632610725,
4.44837874626504781478251437398, 5.37585889208735346786348859705, 6.69220502956559773239392690377, 7.59111921228887151435526646833, 9.235359735038001133082341022135, 10.13071473858409237880329163208, 11.78371664995541256777090216274, 12.91579344036196336152965421658, 14.41943868067291434002100009519, 15.17504411102317779283812417356