| 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 | \( 1 - 14.8iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.388513772145679746426722837876, −9.157856435115486013536635343578, −7.985896817667059940479870803953, −7.47447093172145895430270401050, −6.72996637630738448129875897163, −5.86960166572514984181997014938, −5.20199900037877908056199164458, −4.31392834042464665713594538837, −2.88872793543652654702833825380, −1.88905477895534508055376154293,
0.39029540564452322454680269083, 1.74880183530616904935616320243, 2.77709823792998055210940396600, 3.58267958984426480832226036404, 4.51626160158364359327914286232, 5.62999699156149640233807671369, 6.50401882888075287433743111109, 7.51269387809278510023704412646, 8.207018861660650334661164819729, 9.153429493675716041371549902558
