L(s) = 1 | + (3.04 − 0.814i)3-s + (−1.87 − 0.501i)5-s + (1.89 + 1.84i)7-s + (5.98 − 3.45i)9-s + (−0.299 − 1.11i)11-s + (0.00680 + 0.00680i)13-s − 6.10·15-s + (1.52 − 2.63i)17-s + (−1.24 + 4.64i)19-s + (7.27 + 4.05i)21-s + (4.27 − 2.46i)23-s + (−1.07 − 0.620i)25-s + (8.70 − 8.70i)27-s + (−1.45 − 1.45i)29-s + (−2.60 + 4.51i)31-s + ⋯ |
L(s) = 1 | + (1.75 − 0.470i)3-s + (−0.837 − 0.224i)5-s + (0.717 + 0.696i)7-s + (1.99 − 1.15i)9-s + (−0.0904 − 0.337i)11-s + (0.00188 + 0.00188i)13-s − 1.57·15-s + (0.369 − 0.639i)17-s + (−0.285 + 1.06i)19-s + (1.58 + 0.885i)21-s + (0.891 − 0.514i)23-s + (−0.214 − 0.124i)25-s + (1.67 − 1.67i)27-s + (−0.270 − 0.270i)29-s + (−0.468 + 0.811i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.24131 - 0.515865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.24131 - 0.515865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.89 - 1.84i)T \) |
good | 3 | \( 1 + (-3.04 + 0.814i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (1.87 + 0.501i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (0.299 + 1.11i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.00680 - 0.00680i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.52 + 2.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.24 - 4.64i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-4.27 + 2.46i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.45 + 1.45i)T + 29iT^{2} \) |
| 31 | \( 1 + (2.60 - 4.51i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.51 + 2.28i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 6.02iT - 41T^{2} \) |
| 43 | \( 1 + (7.17 - 7.17i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.796 + 1.38i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.211 - 0.787i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.94 + 7.25i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.46 + 9.21i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (6.35 - 1.70i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 6.77iT - 71T^{2} \) |
| 73 | \( 1 + (-3.43 - 1.98i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.81 - 4.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.59 - 2.07i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.390T + 97T^{2} \) |
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
−11.13824514937263132894363831108, −9.827647656164683509103728521829, −8.908495691018604529670353413259, −8.217553061336960711770271358698, −7.81352653069628428531833261093, −6.71112856642384457085728775191, −5.08444115580718246360541307339, −3.83548450154784048759993172900, −2.90323420552293654864399361193, −1.63030634534452990899188597291,
1.88977193047568663508127765222, 3.31572943911225651534407238345, 4.00575284572855491466251375769, 4.98809479320231650097439354429, 7.15339481958627686809144299733, 7.55405796888908049776469793306, 8.477440688712784140031037186662, 9.138862729679573249921088226405, 10.25608255948070625185263577584, 10.89958039662295909462428368562