L(s) = 1 | + (−1.16 − 0.799i)2-s + (0.721 + 1.86i)4-s + (−3.44 + 1.99i)5-s + (−0.212 − 2.63i)7-s + (0.649 − 2.75i)8-s + (5.61 + 0.434i)10-s + (0.936 − 1.62i)11-s + 1.05·13-s + (−1.86 + 3.24i)14-s + (−2.95 + 2.69i)16-s + (−2.30 − 1.33i)17-s + (−0.628 + 0.362i)19-s + (−6.20 − 4.99i)20-s + (−2.38 + 1.14i)22-s + (3.00 + 5.20i)23-s + ⋯ |
L(s) = 1 | + (−0.824 − 0.565i)2-s + (0.360 + 0.932i)4-s + (−1.54 + 0.890i)5-s + (−0.0804 − 0.996i)7-s + (0.229 − 0.973i)8-s + (1.77 + 0.137i)10-s + (0.282 − 0.489i)11-s + 0.293·13-s + (−0.497 + 0.867i)14-s + (−0.739 + 0.673i)16-s + (−0.558 − 0.322i)17-s + (−0.144 + 0.0831i)19-s + (−1.38 − 1.11i)20-s + (−0.509 + 0.243i)22-s + (0.626 + 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.626230 + 0.122471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.626230 + 0.122471i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.16 + 0.799i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.212 + 2.63i)T \) |
good | 5 | \( 1 + (3.44 - 1.99i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.936 + 1.62i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + (2.30 + 1.33i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.628 - 0.362i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.00 - 5.20i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.09iT - 29T^{2} \) |
| 31 | \( 1 + (-3.48 - 2.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.48 - 7.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.19iT - 41T^{2} \) |
| 43 | \( 1 - 12.3iT - 43T^{2} \) |
| 47 | \( 1 + (-2.03 - 3.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.2 - 6.48i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.56 + 7.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.21 + 9.02i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.00 - 2.88i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + (7.37 - 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.92 + 2.84i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.27T + 83T^{2} \) |
| 89 | \( 1 + (8.80 - 5.08i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.03T + 97T^{2} \) |
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
−10.61766624016561297557103795935, −9.666541780970617520704377404310, −8.578712020829896144633680553913, −7.86217639260329871361835045438, −7.13364734154313541737449201977, −6.54832681232068900429730643763, −4.48460976274893656747149060228, −3.62545996349933794896231624459, −2.96125673957118916056956306760, −0.967317551725543774539805430958,
0.58131543496717197226069702151, 2.30738360745783761244414887258, 4.05932347909148788647222590733, 4.91829263285016328108259618258, 5.99826995359248838707572399556, 7.02790595633328052084289171423, 7.87725411756555231563357092587, 8.769930494382932307891309167968, 8.884002803772410389964502876192, 10.13751483712690428895764233907