L(s) = 1 | + 2.71·2-s + 5.37·4-s + (0.793 − 1.37i)5-s + 9.15·8-s + (2.15 − 3.73i)10-s + (−0.674 − 1.16i)11-s + (−1.58 − 2.75i)13-s + 14.1·16-s + (−1.40 + 2.42i)17-s + (0.312 + 0.541i)19-s + (4.26 − 7.38i)20-s + (−1.83 − 3.17i)22-s + (−0.142 + 0.246i)23-s + (1.24 + 2.15i)25-s + (−4.31 − 7.47i)26-s + ⋯ |
L(s) = 1 | + 1.91·2-s + 2.68·4-s + (0.354 − 0.614i)5-s + 3.23·8-s + (0.681 − 1.17i)10-s + (−0.203 − 0.352i)11-s + (−0.440 − 0.763i)13-s + 3.52·16-s + (−0.339 + 0.588i)17-s + (0.0717 + 0.124i)19-s + (0.952 − 1.65i)20-s + (−0.390 − 0.676i)22-s + (−0.0296 + 0.0514i)23-s + (0.248 + 0.430i)25-s + (−0.846 − 1.46i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.775388377\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.775388377\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 + (-0.793 + 1.37i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.674 + 1.16i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 + 2.75i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.40 - 2.42i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.312 - 0.541i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.142 - 0.246i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.27 - 3.93i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + (4.01 + 6.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.01 - 8.68i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.12 - 5.42i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + (-1.39 + 2.41i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 4.57T + 59T^{2} \) |
| 61 | \( 1 - 0.385T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 1.45T + 71T^{2} \) |
| 73 | \( 1 + (0.234 - 0.405i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + (6.99 - 12.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.29 - 2.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.22 - 12.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.781813323372760737412952462576, −8.641488723365848344680675894985, −7.59138311494122528837877929231, −6.89106249593513777858814099237, −5.66723938166328624509989308836, −5.53608062271600829162965513029, −4.51133946290765784068835199961, −3.62533266682646071135486966181, −2.68815936584588361201428699875, −1.54999343004344067619727349463,
2.00881631329036850639443723724, 2.64728290001652679495617180626, 3.75611096818138388262681882561, 4.55803954319218364358114725395, 5.40526731138734261820223586949, 6.16858518364425064831901601410, 7.10545550538620617656099991586, 7.34303310023645735572746968410, 8.894516017470814346083459512686, 10.07241278847632842799612493537