L(s) = 1 | + (−1.42 + 1.42i)3-s + (0.707 + 0.707i)5-s − 0.690i·7-s − 1.05i·9-s + (3.06 + 3.06i)11-s + (−2.33 + 2.33i)13-s − 2.01·15-s − 5.28·17-s + (−5.38 + 5.38i)19-s + (0.982 + 0.982i)21-s − 1.60i·23-s + 1.00i·25-s + (−2.77 − 2.77i)27-s + (1.70 − 1.70i)29-s + 4.69·31-s + ⋯ |
L(s) = 1 | + (−0.821 + 0.821i)3-s + (0.316 + 0.316i)5-s − 0.261i·7-s − 0.350i·9-s + (0.922 + 0.922i)11-s + (−0.648 + 0.648i)13-s − 0.519·15-s − 1.28·17-s + (−1.23 + 1.23i)19-s + (0.214 + 0.214i)21-s − 0.335i·23-s + 0.200i·25-s + (−0.533 − 0.533i)27-s + (0.316 − 0.316i)29-s + 0.843·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.526 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.424081 + 0.761910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.424081 + 0.761910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.42 - 1.42i)T - 3iT^{2} \) |
| 7 | \( 1 + 0.690iT - 7T^{2} \) |
| 11 | \( 1 + (-3.06 - 3.06i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.33 - 2.33i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 19 | \( 1 + (5.38 - 5.38i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.60iT - 23T^{2} \) |
| 29 | \( 1 + (-1.70 + 1.70i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 + (-7.89 - 7.89i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.49iT - 41T^{2} \) |
| 43 | \( 1 + (-0.256 - 0.256i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + (4.99 + 4.99i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.46 + 1.46i)T + 59iT^{2} \) |
| 61 | \( 1 + (-9.33 + 9.33i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.94 + 1.94i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.32iT - 71T^{2} \) |
| 73 | \( 1 - 1.29iT - 73T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + (7.30 - 7.30i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.81iT - 89T^{2} \) |
| 97 | \( 1 - 5.27T + 97T^{2} \) |
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
−11.74269809930098898337004808975, −10.98834932059017488803063549614, −10.05763663875540451251409409573, −9.564224811983077658741673786611, −8.233072541620197969481049018525, −6.78646383076683245982297636177, −6.18781222090406220800581704191, −4.66351646916684599909583796422, −4.19125258789816507061716759531, −2.13459477908074309700878248101,
0.68778197718855885716396499145, 2.42976650596328121545957229978, 4.30305451264319162722638082401, 5.61535487839644832328335849054, 6.37624416020275519731105524261, 7.19655824286704490844259133671, 8.614020163414144474268430395289, 9.254110963542792780019326323134, 10.70922321845861482604586514360, 11.39398701678973372604630602628