L(s) = 1 | + (−1.36 + 0.787i)3-s + (0.476 + 0.274i)5-s + (2.60 + 0.447i)7-s + (−0.258 + 0.447i)9-s + (−2.07 + 1.19i)11-s + 3.96i·13-s − 0.866·15-s + (2.10 + 3.65i)17-s + (5.75 + 3.32i)19-s + (−3.91 + 1.44i)21-s + (−1.17 + 2.03i)23-s + (−2.34 − 4.06i)25-s − 5.54i·27-s − 8.21i·29-s + (0.433 + 0.750i)31-s + ⋯ |
L(s) = 1 | + (−0.787 + 0.454i)3-s + (0.212 + 0.122i)5-s + (0.985 + 0.169i)7-s + (−0.0862 + 0.149i)9-s + (−0.624 + 0.360i)11-s + 1.10i·13-s − 0.223·15-s + (0.511 + 0.885i)17-s + (1.31 + 0.761i)19-s + (−0.853 + 0.314i)21-s + (−0.244 + 0.424i)23-s + (−0.469 − 0.813i)25-s − 1.06i·27-s − 1.52i·29-s + (0.0777 + 0.134i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.807965 + 0.593485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.807965 + 0.593485i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.60 - 0.447i)T \) |
good | 3 | \( 1 + (1.36 - 0.787i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.476 - 0.274i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.07 - 1.19i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.96iT - 13T^{2} \) |
| 17 | \( 1 + (-2.10 - 3.65i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.75 - 3.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.17 - 2.03i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.21iT - 29T^{2} \) |
| 31 | \( 1 + (-0.433 - 0.750i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.229 - 0.132i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 5.35iT - 43T^{2} \) |
| 47 | \( 1 + (-1.29 + 2.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.36 + 5.40i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.26 - 1.88i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.18 - 3.56i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.31 + 1.33i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.76T + 71T^{2} \) |
| 73 | \( 1 + (2.33 + 4.03i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.308 + 0.533i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.09iT - 83T^{2} \) |
| 89 | \( 1 + (-3.19 + 5.53i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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
−11.88005777134619101477308417613, −11.73464943478457400692784745387, −10.45515278626296344780350567583, −9.886980059083760942884168423599, −8.414835997009059233102767897026, −7.52609220771178894104623131004, −5.99688122282491416004114052683, −5.21421840380751646515813043316, −4.10306270203651159561723848876, −2.01710376048407178105703777968,
1.00603095853186966520482077326, 3.09044302105114976658539421255, 5.12270071083295227513030617244, 5.56242162067925312582556354663, 7.06885921953238991620137557111, 7.892744996491355814738290015887, 9.100028361806781398305001666386, 10.35684139718886961973014861964, 11.26075221045713715634650093169, 11.91419137982437460372532976595