| L(s) = 1 | + (−0.727 − 0.807i)2-s + (−0.917 − 1.46i)3-s + (0.0855 − 0.813i)4-s + (0.934 − 2.03i)5-s + (−0.519 + 1.80i)6-s + (1.73 − 3.01i)7-s + (−2.47 + 1.80i)8-s + (−1.31 + 2.69i)9-s + (−2.32 + 0.722i)10-s + (2.35 + 2.61i)11-s + (−1.27 + 0.620i)12-s + (1.45 − 1.61i)13-s + (−3.69 + 0.786i)14-s + (−3.84 + 0.490i)15-s + (1.65 + 0.352i)16-s + (−1.54 + 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.514 − 0.571i)2-s + (−0.529 − 0.848i)3-s + (0.0427 − 0.406i)4-s + (0.417 − 0.908i)5-s + (−0.212 + 0.738i)6-s + (0.657 − 1.13i)7-s + (−0.876 + 0.636i)8-s + (−0.439 + 0.898i)9-s + (−0.733 + 0.228i)10-s + (0.709 + 0.788i)11-s + (−0.367 + 0.179i)12-s + (0.402 − 0.447i)13-s + (−0.988 + 0.210i)14-s + (−0.991 + 0.126i)15-s + (0.414 + 0.0880i)16-s + (−0.375 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0966030 - 0.851467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0966030 - 0.851467i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.917 + 1.46i)T \) |
| 5 | \( 1 + (-0.934 + 2.03i)T \) |
| good | 2 | \( 1 + (0.727 + 0.807i)T + (-0.209 + 1.98i)T^{2} \) |
| 7 | \( 1 + (-1.73 + 3.01i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.35 - 2.61i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.45 + 1.61i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.54 - 1.12i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.970 - 0.704i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.96 - 0.417i)T + (21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (3.26 - 1.45i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-5.79 - 2.58i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.46 - 7.58i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.73 + 1.92i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-5.34 + 9.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-11.9 + 5.34i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (11.2 + 8.18i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.64 - 4.04i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.353 - 0.392i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (9.83 + 4.37i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-6.38 - 4.63i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.75 - 8.46i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.17 + 2.74i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.372 - 3.53i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-2.71 + 8.36i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (15.6 - 6.98i)T + (64.9 - 72.0i)T^{2} \) |
| 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
−11.76865463406391166206515741429, −10.78596986566261388129312930720, −10.09971252149568245644673899674, −8.890703104325643878203167592887, −7.932468704386456032129577694301, −6.68382813077524476658892387748, −5.59264648539826489941502259288, −4.43695632030613276534851291474, −1.90477935073733786095969733901, −0.971763339454350420492917129443,
2.78559570253842474530398484169, 4.17823627169106997356117649982, 5.93015416532149255749269629357, 6.31866046348892520088068436004, 7.80577669542527337066025113444, 9.029548192692308482652555016022, 9.392272836234706401852527372940, 10.93684691554729500810331688312, 11.46902159999408117311117898626, 12.36238202482677845519397460675
