L(s) = 1 | + (1 − 1.73i)4-s + 7·13-s + (−1.99 − 3.46i)16-s + (−3.5 − 6.06i)19-s + (2.5 − 4.33i)25-s + (−3.5 + 6.06i)31-s + (0.5 + 0.866i)37-s + 5·43-s + (7 − 12.1i)52-s + (7 + 12.1i)61-s − 7.99·64-s + (−5.5 + 9.52i)67-s + (−3.5 + 6.06i)73-s − 14·76-s + (6.5 + 11.2i)79-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + 1.94·13-s + (−0.499 − 0.866i)16-s + (−0.802 − 1.39i)19-s + (0.5 − 0.866i)25-s + (−0.628 + 1.08i)31-s + (0.0821 + 0.142i)37-s + 0.762·43-s + (0.970 − 1.68i)52-s + (0.896 + 1.55i)61-s − 0.999·64-s + (−0.671 + 1.16i)67-s + (−0.409 + 0.709i)73-s − 1.60·76-s + (0.731 + 1.26i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43315 - 0.710455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43315 - 0.710455i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 7T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.90353487336046868288197811939, −10.36540667070145818476042774431, −9.103973899798186914167211511003, −8.467170440399022191272434916402, −7.00540619073477057371705526330, −6.33877706044315874331776533094, −5.39149127483252496338818142677, −4.15021255882218326662270793415, −2.64949038910737600429711607958, −1.15061253936147325347589354896,
1.78678407936487485488284490821, 3.36028575022800212308720570501, 4.09192514111419180474122829763, 5.79882880499407263774288320665, 6.55084563165038723035480240860, 7.72728847174516648442637002954, 8.397336184619730793573480375057, 9.283608278934972193511150123357, 10.70602380960564718246505483983, 11.12860371615261467851924439168