L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.64i·7-s + 2.82·8-s + 5.15i·11-s + 6.04i·13-s − 3.74i·14-s + 4.00·16-s − 5.99·17-s + 10.0·19-s + 7.29i·22-s + 20.1·23-s + 8.54i·26-s − 5.29i·28-s − 26.5i·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.377i·7-s + 0.353·8-s + 0.468i·11-s + 0.464i·13-s − 0.267i·14-s + 0.250·16-s − 0.352·17-s + 0.529·19-s + 0.331i·22-s + 0.874·23-s + 0.328i·26-s − 0.188i·28-s − 0.915i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.455642232\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.455642232\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 11 | \( 1 - 5.15iT - 121T^{2} \) |
| 13 | \( 1 - 6.04iT - 169T^{2} \) |
| 17 | \( 1 + 5.99T + 289T^{2} \) |
| 19 | \( 1 - 10.0T + 361T^{2} \) |
| 23 | \( 1 - 20.1T + 529T^{2} \) |
| 29 | \( 1 + 26.5iT - 841T^{2} \) |
| 31 | \( 1 + 6.40T + 961T^{2} \) |
| 37 | \( 1 - 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 54.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.37iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 20.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 48.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 60.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 35.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 85.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 42.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 35.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 49.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 17.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 123. iT - 9.40e3T^{2} \) |
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
−8.460226172521346337960029020090, −7.66179294569049952625165897269, −6.93149437662219135284493105311, −6.37413964094622195427729210119, −5.37949462934930477223685207782, −4.66400969462114915443453714657, −3.97186459968967214062141465062, −3.03076762184683620770481321268, −2.09246326446867407013151709166, −0.964948303483395842733652594539,
0.69884719869120372494781979863, 1.98103462444741394409555064038, 2.95342116747963285705759325132, 3.64250798566413256452165909385, 4.64668752617524107664764773614, 5.47468200926484799910933141419, 5.91339088944093174371862658946, 7.01231909251524389948167155576, 7.43580034544378489199034440148, 8.579365555642615392340326639024