L(s) = 1 | + 3-s + 2.90·5-s + (−0.217 − 0.376i)7-s + 9-s + (2.53 + 4.39i)11-s + (−1.58 + 2.74i)13-s + 2.90·15-s + (0.150 − 0.260i)17-s + (−1.17 + 2.02i)19-s + (−0.217 − 0.376i)21-s + (1.5 − 2.59i)23-s + 3.46·25-s + 27-s + (−2.95 − 5.11i)29-s + (0.937 + 1.62i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.30·5-s + (−0.0821 − 0.142i)7-s + 0.333·9-s + (0.765 + 1.32i)11-s + (−0.440 + 0.762i)13-s + 0.751·15-s + (0.0364 − 0.0631i)17-s + (−0.268 + 0.465i)19-s + (−0.0474 − 0.0821i)21-s + (0.312 − 0.541i)23-s + 0.693·25-s + 0.192·27-s + (−0.548 − 0.950i)29-s + (0.168 + 0.291i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.32886 + 0.407634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.32886 + 0.407634i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (-8.16 - 0.601i)T \) |
good | 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 + (0.217 + 0.376i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.53 - 4.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 - 2.74i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.150 + 0.260i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.17 - 2.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.95 + 5.11i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.937 - 1.62i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.62 + 8.00i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.18 + 2.05i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 + (-3.88 - 6.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + (-3.82 + 6.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (3.57 + 6.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.22 - 10.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.188 - 0.325i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.62 + 2.81i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + (0.127 - 0.220i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.891247376291077499192964837682, −9.608970587551151609992614288386, −8.886854641333555134699757831451, −7.66230964321839004691689890792, −6.81850242040286476980113325583, −6.06352786563344511178295503838, −4.83096525069595337907522281480, −3.95417035603408264771270861149, −2.39876506677986799349090875808, −1.70626605116071304402635488218,
1.30530155213276258816773888159, 2.62515591209729302214247745263, 3.49409570345034926835385128310, 4.97016155828957292396104573928, 5.88655492754599426508616034665, 6.57912075097945349259843419141, 7.75398540147672816215331786272, 8.733004445445717735343922530676, 9.302662114743735793762891040321, 10.05917551293099183634860312348