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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.413632507\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.413632507\) |
\(L(\frac{3}{2})\) |
|
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
−9.986265874333401967713952993914, −8.343472250911054594536866299073, −7.88328174599132213049814622916, −6.71628917331921606613506795055, −6.08755640165641020234276855664, −5.13453953893549705134623390215, −4.56270438474160822324911113309, −3.64285584160303480059317516075, −2.79400896479495176889038590603, −1.46978050315112993906775463550,
1.78064299641430404417096867330, 3.01377821699918058107472484657, 3.55103531230910929374242361563, 4.56578363571083494107359997692, 5.34165296277938146626341796130, 6.25225746847995865760155882885, 6.94361420938292663088405470394, 7.55706724943286174548942689571, 8.724133619094491420270373314651, 10.06001285125834674934927741816