L(s) = 1 | + (−0.5 + 0.866i)2-s + (3.5 + 6.06i)4-s + (−6 + 10.3i)5-s − 15·8-s + (−6 − 10.3i)10-s + (10 + 17.3i)11-s + 84·13-s + (−20.5 + 35.5i)16-s + (48 + 83.1i)17-s + (6 − 10.3i)19-s − 84·20-s − 20·22-s + (−88 + 152. i)23-s + (−9.5 − 16.4i)25-s + (−42 + 72.7i)26-s + ⋯ |
L(s) = 1 | + (−0.176 + 0.306i)2-s + (0.437 + 0.757i)4-s + (−0.536 + 0.929i)5-s − 0.662·8-s + (−0.189 − 0.328i)10-s + (0.274 + 0.474i)11-s + 1.79·13-s + (−0.320 + 0.554i)16-s + (0.684 + 1.18i)17-s + (0.0724 − 0.125i)19-s − 0.939·20-s − 0.193·22-s + (−0.797 + 1.38i)23-s + (−0.0759 − 0.131i)25-s + (−0.316 + 0.548i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.494232607\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.494232607\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (6 - 10.3i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-10 - 17.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 84T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-48 - 83.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-6 + 10.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (88 - 152. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 58T + 2.43e4T^{2} \) |
| 31 | \( 1 + (132 + 228. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (129 - 223. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 156T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-204 + 353. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (361 + 625. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (246 + 426. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (246 - 426. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (206 + 356. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 296T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-120 - 207. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (388 - 672. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 924T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-372 + 644. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 168T + 9.12e5T^{2} \) |
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
−11.26944873593136688638968412223, −10.39507997126225145771758139324, −9.175128984338288616165063575190, −8.133220958571307050313269714085, −7.56152412251260532234993786401, −6.57476235179646911101079991341, −5.82736808873831508691561200318, −3.76933066749669812179729223869, −3.48641735823991290355648251467, −1.79590207195838378719183194059,
0.53682872322282660209933771217, 1.44176641217406390349784094451, 3.11590869055043645064590088894, 4.36641492013872712747760207101, 5.57144650156254054005810475612, 6.33001992566556794931794026253, 7.60246869281686540254680720112, 8.759274460311659251883494434803, 9.168505813606015775572158865010, 10.51580462255697629261295100852