L(s) = 1 | + (−13.4 + 13.4i)3-s + (15.0 + 53.8i)5-s + (−76.4 − 76.4i)7-s − 117. i·9-s − 622. i·11-s + (−293. − 293. i)13-s + (−925. − 521. i)15-s + (−1.15e3 + 1.15e3i)17-s + 2.00e3·19-s + 2.05e3·21-s + (1.39e3 − 1.39e3i)23-s + (−2.67e3 + 1.61e3i)25-s + (−1.68e3 − 1.68e3i)27-s + 305. i·29-s + 2.10e3i·31-s + ⋯ |
L(s) = 1 | + (−0.861 + 0.861i)3-s + (0.268 + 0.963i)5-s + (−0.589 − 0.589i)7-s − 0.485i·9-s − 1.55i·11-s + (−0.481 − 0.481i)13-s + (−1.06 − 0.598i)15-s + (−0.967 + 0.967i)17-s + 1.27·19-s + 1.01·21-s + (0.548 − 0.548i)23-s + (−0.855 + 0.518i)25-s + (−0.443 − 0.443i)27-s + 0.0675i·29-s + 0.393i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9786008140\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9786008140\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-15.0 - 53.8i)T \) |
good | 3 | \( 1 + (13.4 - 13.4i)T - 243iT^{2} \) |
| 7 | \( 1 + (76.4 + 76.4i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 622. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (293. + 293. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.15e3 - 1.15e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.39e3 + 1.39e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 305. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.10e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-9.90e3 + 9.90e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.69e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (2.03e3 - 2.03e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (682. + 682. i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-2.10e4 - 2.10e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 1.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.30e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.98e4 + 3.98e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.54e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.03e3 - 1.03e3i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 1.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-4.51e4 + 4.51e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.43e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (2.43e4 - 2.43e4i)T - 8.58e9iT^{2} \) |
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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
−11.49900793963599113151962354389, −10.79402301252043015901055280816, −10.29372868765482382970104446999, −9.157575616566461149869816999628, −7.58746482695090111059575396209, −6.32222839660868980281701767486, −5.56744056517739819650152462003, −4.04431662949481882812070159768, −2.89753591575533337412428868412, −0.45600610741081031681833466976,
0.976546217881951214578717580638, 2.32772902192153018731103182030, 4.59391223303572748895709203704, 5.54487240099500019290296687982, 6.71655081371207418400240391958, 7.54937025366068494358929476067, 9.272891033793494380059541522810, 9.662113975780149760413552153754, 11.53687890127420008059797494091, 12.05710589317661512099450594191