L(s) = 1 | + (−0.159 + 0.200i)2-s + (0.430 + 1.88i)4-s + (1.50 + 0.726i)5-s + (2.63 − 0.216i)7-s + (−0.909 − 0.437i)8-s + (−0.387 + 0.186i)10-s + (2.04 − 2.56i)11-s + (1.69 − 2.12i)13-s + (−0.378 + 0.563i)14-s + (−3.25 + 1.56i)16-s + (−0.947 + 4.15i)17-s − 2.59·19-s + (−0.721 + 3.15i)20-s + (0.187 + 0.819i)22-s + (1.32 + 5.82i)23-s + ⋯ |
L(s) = 1 | + (−0.113 + 0.141i)2-s + (0.215 + 0.942i)4-s + (0.675 + 0.325i)5-s + (0.996 − 0.0819i)7-s + (−0.321 − 0.154i)8-s + (−0.122 + 0.0589i)10-s + (0.616 − 0.772i)11-s + (0.469 − 0.589i)13-s + (−0.101 + 0.150i)14-s + (−0.813 + 0.391i)16-s + (−0.229 + 1.00i)17-s − 0.595·19-s + (−0.161 + 0.706i)20-s + (0.0398 + 0.174i)22-s + (0.277 + 1.21i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50903 + 0.760397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50903 + 0.760397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.216i)T \) |
good | 2 | \( 1 + (0.159 - 0.200i)T + (-0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-1.50 - 0.726i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.04 + 2.56i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.69 + 2.12i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.947 - 4.15i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 + (-1.32 - 5.82i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.403 + 1.76i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + (-0.755 + 3.31i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-5.05 - 2.43i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (9.10 - 4.38i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (0.656 - 0.823i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (2.01 + 8.82i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (10.7 - 5.18i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (1.71 - 7.49i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + (3.66 + 16.0i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.59 + 5.76i)T + (-16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + 0.219T + 79T^{2} \) |
| 83 | \( 1 + (10.0 + 12.6i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-10.7 - 13.5i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23405026749006623284773377990, −10.55092863597117331215820079161, −9.278836374800547106387892907727, −8.363156010118297458625428731536, −7.80263409326595651530333505310, −6.53816439271316012114168447004, −5.79526366690247253470800866849, −4.28944879112150927488551508266, −3.21450709404782154983340105201, −1.77842190616300638990350274413,
1.36192404063387251167958798133, 2.30034443111729673725886692110, 4.43595518680061121222851504556, 5.15949704222186736660058308202, 6.27798221066853358739467057820, 7.09269318233834088891579997808, 8.555536681347300507023497996124, 9.261707999567479864409747046242, 10.04992118915863526454444579788, 11.01793905160202946203085789274