L(s) = 1 | − 2i·2-s − 1.74i·3-s − 4·4-s + (4.87 − 1.12i)5-s − 3.48·6-s − 7i·7-s + 8i·8-s + 5.96·9-s + (−2.25 − 9.74i)10-s + 6.96i·12-s − 21.7i·13-s − 14·14-s + (−1.96 − 8.48i)15-s + 16·16-s − 11.9i·18-s − 32.1·19-s + ⋯ |
L(s) = 1 | − i·2-s − 0.580i·3-s − 4-s + (0.974 − 0.225i)5-s − 0.580·6-s − i·7-s + i·8-s + 0.662·9-s + (−0.225 − 0.974i)10-s + 0.580i·12-s − 1.67i·13-s − 14-s + (−0.131 − 0.565i)15-s + 16-s − 0.662i·18-s − 1.69·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.184070 - 1.60908i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.184070 - 1.60908i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 + (-4.87 + 1.12i)T \) |
| 7 | \( 1 + 7iT \) |
good | 3 | \( 1 + 1.74iT - 9T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 21.7iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 32.1T + 361T^{2} \) |
| 23 | \( 1 - 44.8iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 9.60T + 3.48e3T^{2} \) |
| 61 | \( 1 - 80.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 89.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 89.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 133. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.01415975873991121332924284267, −10.28063015366404595922395370274, −9.688466632619160779950697324838, −8.388380105245402626473496476356, −7.42750663496746075811170641854, −6.05536527119852521981148437088, −4.88243964759236697288290498892, −3.57392114269976444739407280183, −2.02384707449424628895067689616, −0.858832909094274798254945025081,
2.14710593620478366639738313363, 4.16442733809072998996234787985, 5.01815115087204119561256123227, 6.33411948958469444269447632467, 6.75189554850861888903195968609, 8.485800754009883607946744035657, 9.121044808070631694747501776551, 9.895777469369921122403655766622, 10.84098135025982081402487360976, 12.38204908911474846582402321496