L(s) = 1 | + (−0.720 − 0.720i)3-s + 4.02·7-s − 1.96i·9-s + (0.646 + 0.646i)11-s + (−4.91 − 4.91i)13-s − 2.70i·17-s + (−0.438 + 0.438i)19-s + (−2.90 − 2.90i)21-s + 3.60·23-s + (−3.57 + 3.57i)27-s + (−2.00 + 2.00i)29-s − 4.30·31-s − 0.932i·33-s + (0.743 − 0.743i)37-s + 7.08i·39-s + ⋯ |
L(s) = 1 | + (−0.416 − 0.416i)3-s + 1.52·7-s − 0.653i·9-s + (0.195 + 0.195i)11-s + (−1.36 − 1.36i)13-s − 0.656i·17-s + (−0.100 + 0.100i)19-s + (−0.633 − 0.633i)21-s + 0.750·23-s + (−0.688 + 0.688i)27-s + (−0.373 + 0.373i)29-s − 0.774·31-s − 0.162i·33-s + (0.122 − 0.122i)37-s + 1.13i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.365925104\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.365925104\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.720 + 0.720i)T + 3iT^{2} \) |
| 7 | \( 1 - 4.02T + 7T^{2} \) |
| 11 | \( 1 + (-0.646 - 0.646i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.91 + 4.91i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.70iT - 17T^{2} \) |
| 19 | \( 1 + (0.438 - 0.438i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 + (2.00 - 2.00i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + (-0.743 + 0.743i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.603iT - 41T^{2} \) |
| 43 | \( 1 + (-5.03 + 5.03i)T - 43iT^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 + (-4.07 + 4.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.22 - 1.22i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.98 - 6.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.24 + 5.24i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.7iT - 71T^{2} \) |
| 73 | \( 1 - 1.30T + 73T^{2} \) |
| 79 | \( 1 - 0.611T + 79T^{2} \) |
| 83 | \( 1 + (-1.29 - 1.29i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 + 12.7iT - 97T^{2} \) |
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
−9.099837934456953161276462530035, −8.292828391360531595812149814883, −7.32383050064170793378994042515, −7.10164954178311106922398836389, −5.59831361365197006357315373520, −5.25696097529044943617531901947, −4.25004672047869875532254905790, −2.96768067814035821069045525472, −1.79250783913226421021450931265, −0.56217919923992016736836681567,
1.57072912303703601185959927116, 2.44402230617482149446809609832, 4.12878627310322997613896275666, 4.69877630774113836507035528961, 5.31254472119844017544228790929, 6.36637424653607665958891127087, 7.48715034670512315922463076955, 7.895864222830785574130199415691, 8.985186729390401042464518895601, 9.568918570942707602204310926905