L(s) = 1 | − 1.56i·2-s − 0.438·4-s − i·7-s − 2.43i·8-s − 2.56·11-s − 4.56i·13-s − 1.56·14-s − 4.68·16-s + 4.56i·17-s − 1.12·19-s + 4i·22-s − 5.12i·23-s − 7.12·26-s + 0.438i·28-s − 5.68·29-s + ⋯ |
L(s) = 1 | − 1.10i·2-s − 0.219·4-s − 0.377i·7-s − 0.862i·8-s − 0.772·11-s − 1.26i·13-s − 0.417·14-s − 1.17·16-s + 1.10i·17-s − 0.257·19-s + 0.852i·22-s − 1.06i·23-s − 1.39·26-s + 0.0828i·28-s − 1.05·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.090060487\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.090060487\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 2 | \( 1 + 1.56iT - 2T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.56iT - 13T^{2} \) |
| 17 | \( 1 - 4.56iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 + 9.12iT - 43T^{2} \) |
| 47 | \( 1 + 3.68iT - 47T^{2} \) |
| 53 | \( 1 - 3.12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 + 6.24iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4.24iT - 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 + 14.8iT - 97T^{2} \) |
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
−9.153429493675716041371549902558, −8.207018861660650334661164819729, −7.51269387809278510023704412646, −6.50401882888075287433743111109, −5.62999699156149640233807671369, −4.51626160158364359327914286232, −3.58267958984426480832226036404, −2.77709823792998055210940396600, −1.74880183530616904935616320243, −0.39029540564452322454680269083,
1.88905477895534508055376154293, 2.88872793543652654702833825380, 4.31392834042464665713594538837, 5.20199900037877908056199164458, 5.86960166572514984181997014938, 6.72996637630738448129875897163, 7.47447093172145895430270401050, 7.985896817667059940479870803953, 9.157856435115486013536635343578, 9.388513772145679746426722837876