| L(s) = 1 | + (0.743 − 0.669i)2-s + (0.679 + 1.59i)3-s + (0.104 − 0.994i)4-s + (0.798 + 2.08i)5-s + (1.57 + 0.728i)6-s + (2.11 + 1.21i)7-s + (−0.587 − 0.809i)8-s + (−2.07 + 2.16i)9-s + (1.99 + 1.01i)10-s + (0.211 + 0.234i)11-s + (1.65 − 0.509i)12-s + (−4.38 − 3.95i)13-s + (2.38 − 0.507i)14-s + (−2.78 + 2.69i)15-s + (−0.978 − 0.207i)16-s + (3.64 + 5.01i)17-s + ⋯ |
| L(s) = 1 | + (0.525 − 0.473i)2-s + (0.392 + 0.919i)3-s + (0.0522 − 0.497i)4-s + (0.356 + 0.934i)5-s + (0.641 + 0.297i)6-s + (0.798 + 0.460i)7-s + (−0.207 − 0.286i)8-s + (−0.691 + 0.722i)9-s + (0.629 + 0.322i)10-s + (0.0637 + 0.0708i)11-s + (0.477 − 0.147i)12-s + (−1.21 − 1.09i)13-s + (0.637 − 0.135i)14-s + (−0.719 + 0.694i)15-s + (−0.244 − 0.0519i)16-s + (0.883 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.05295 + 0.838880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05295 + 0.838880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.743 + 0.669i)T \) |
| 3 | \( 1 + (-0.679 - 1.59i)T \) |
| 5 | \( 1 + (-0.798 - 2.08i)T \) |
| good | 7 | \( 1 + (-2.11 - 1.21i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.211 - 0.234i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (4.38 + 3.95i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-3.64 - 5.01i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.706 - 0.513i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.01 + 4.78i)T + (-21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-8.57 + 3.81i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-2.30 - 1.02i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-3.71 + 1.20i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.44 + 3.82i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.81 - 1.62i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.16 + 7.11i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (3.18 - 4.38i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.70 + 1.88i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (6.20 + 6.89i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.119 - 0.269i)T + (-44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 8.18i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.80 + 2.86i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (13.4 - 5.96i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.24 - 0.130i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-4.73 + 14.5i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.09 + 2.44i)T + (-64.9 + 72.0i)T^{2} \) |
| 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
−11.02757259039552214224909813743, −10.20060557737958863028618909122, −9.959281807068018503076786795321, −8.509294431881055518779412388356, −7.72692954043390994934178759468, −6.18282171533770455544004618685, −5.31029620032885864998092748001, −4.33048661135735954980737606994, −3.06306350819034172171287044336, −2.25644677202684363097321546988,
1.31116224589345630609373135940, 2.72571207624468589274365620994, 4.42020004433785534925653482106, 5.16871816706686938927864086260, 6.36921547815263432943550025901, 7.40457666078113313115419060963, 7.947565948954483088615346867833, 9.023265781385794477046975438015, 9.770284554731779478800780064550, 11.48944056630536768118929536744
